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AN XRAY SCATTERING STUDY OF ORDERING IN BLOCK COPOLYMERS By CURTIS RAY HARKLESS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1990 ACKNOWLEDGMENTS Firstly, I would like to express my gratitude to my advisor, Stephen E. Nagler. His assistance in the preparation of this dissertation is greatly appreciated, and without his supervision this work would not have been possible. I would like to express my thanks to the beamline staff, Brian Stephenson and Jean JordanSweet, for their valuable technical assistance and to DARPA and DOE for financial support in the early and later stages of this work, respectively. Much of this work could not have been performed without the outstanding technical support offered by Ward Ruby. His creativity and resourcefulness were second only to his jokes of questionable taste. I thank Paul Lyman and Liz Seiberling not only for their helpful discussions and honest opinions, but also for late night support and indefatigable good humor that I could always count on. I would like to thank Robert F. Shannon, Jr., for all of his help. Writing this dissertation was made a great deal easier by his company and the knowledge that someone else was suffering as much as I. The assistance given to me by Marsha A. Singh in all aspects of the performance of these experiments cannot be overstated. Our trip to Brookhaven to take the data, upon which this dissertation is based, was made less arduous by her companionship and experimental expertise. In addition, her attention to detail was invaluable during the writing of this dissertation. I thank the Lord for supporting me through this challenging time. Finally, I thank my family. The advice offered by my parents: Les and Nancy (like "Get a job." ) and the stimulating questions posed by my sisters: Lisa, Kim, Jen, and Stefanie (like "Aren't you done yet?" ) were greatly appreciated. But most of all, I am grateful for their unfailing support in all of my endeavors. TABLE OF CONTENTS ACKNOWLEDGEMENTS.............................. ....................................................ii LIST OF FIGU RES .............................................................................................. vii ABSTRACT......................................... .............................................................. xii CHAPTERS 1 INTRODUCTION.............................................................................. 1 2 PHASE TRANSITION KINETICS................................ ........... .... 9 Phase Transition KineticsGeneral Concepts ...................................... 9 Nucleation Theory............................................................................ 17 Spinodal Decomposition ................................................................ 21 Domain Growth ........................................................................ 23 Transformation Curves........................................................... ..... 24 Late Stage GrowthCoarsening .................................... ............ 32 Universal Classification..................................................... .......... 34 3 XRAY DIFFRACTION............................................... ............ 36 Fundamentals of Xray Diffraction................................. .......... 36 Small Angle Xray Scattering............................................................. 41 Scattering From Liquids............................................... ............ 47 Scattering From Ordered Structures .............................................. 50 The Effect of Finite Size...................... .......... ........... ... 52 The Crystal Structure Factor............................. ........... .. 53 The Effect of Particle Motion .............................................. 54 The Effect of Strain ............................................................ 55 The Powder Pattern ........................................... ........... ... 56 4 REVIEW OF BLOCK COPOLYMER LITERATURE...................... 59 PolymersGeneral Background.................................... ........... .... 59 The Glass and Melting Transitions................................................. 61 Phase Separation in Polymer Blends................................ ............ 62 Block CopolymersBulk Properties.................................................... 64 Block CopolymersKinetic Studies .................................................... 69 Block Copolymer Solutions.......................................................... 70 Equilibrium Measurements ................................................... 72 TimeDependent Experiments............................................ 75 5 THE EXPERIMENTAL APPARATUS ............................................ 79 System Overview ........................................................................... 79 The Polymer Xray Scattering Furnace............................................ 82 The Furnace Body .............................................................. 82 The Translational Stage ...................................................... 85 The Sample Mount ............................................................. 86 The Quench Technique............................................................ 89 The Xray Scattering Apparatus ..................................................... 92 The National Synchrotron Light Source.................................. 92 The Collimation System ....................................................... 94 The prehutch collimation system............................... 96 The inhutch collimation system................................. 97 The Detection System .......................................................... 98 The detector operating principle................................. 98 The dark count ......................................................... 99 Preliminary Experimental Procedures............................ .............. 100 Sample Preparation.............. .............................. ........ 100 Xray Beam Alignment and Collimation.............. ........... 105 Normalization for Incident Intensity ................................. 108 Measurement of the Parasitic Scattering ............... ............... 109 Equilibrium Data Acquisition Procedure .................................... 112 Kinetic Data Acquisition Procedure ................................................... 113 6 EQUILIBRIUM MEASUREMENTS: ANALYSIS AND RESULTS.................................................................... 117 Presentation of the Static Data................................ 117 Identification of the Xray Scattering Features .............................. 130 Preliminary Discussion of the Static Data................................ 131 The Effects of Experimental Resolution........................ ........ .. 133 Determination of the Structural Parameters ................................. 134 The Microdomain Parameters................................. ... 134 Analysis of the Macrolattice Structure and Dimensions ....... 137 The background contribution............................ 139 The amorphous contribution........................ 139 The crystalline contribution..................... ........ 140 Discussion of the Macrolattice Characterization..................... 140 Finite Size and Strain Effects............................................... 147 The Ordered Volume Fraction.................. ............. 148 Discussion of the Fine Structure in the SAXS Profiles......... 149 Concentration and Temperature Dependence of the Structure............................................. ............................... 152 Behavior of the Crystalline Component ................................. 154 Behavior of the Amorphous Component.............................. 161 7 KINETIC MEASUREMENTS: ANALYSIS AND RESULTS ........ 168 Preliminary Discussion...................................................................... 168 The Ordering Kinetics................................................................... 174 The Master Curve............................................................. 193 The Crossover Behavior ...................................................... 198 Shallow Quench Oscillatory Behavior ................................... 200 8 SUMMARY AND CONCLUSIONS................................................. 206 APPENDICES A DATA ACQUISITION PROGRAMS ............................................... 209 B THE HEATING CIRCUIT .......................................................... 227 C THE SOLENOID INTERFACE .................................................. 229 D THE DEVELOPMENT OF THE SAMPLE MOUNT....................... 231 REFERENCES .................................................................................................. ... 241 BIOGRAPHICAL SKETCH ............................................................................... 249 LIST OF FIGURES Figure 11 The diblock copolymer molecule and an illustration of microphase separation........................................... .............................. 3 12 The three domain morphologies commonly observed in diblock copolymer system s...................................................................................... 5 13 An illustration of the ordering of spherical microdomains onto a cubic lattice......................................................................................................... 6 21 An example of a phase diagram for a binary alloy exhibiting a phase transition....................................................................................... ............ 10 22 The shape of the free energy curve versus order parameter for various temperatures, above and below the transition point......................................... 13 23 An illustration of the different temporal regimes in the transformation process............................................................. ................................. 16 24 A contrast of the nucleation and spinodal decomposition transformation mechanisms ................................................................................................. 22 25 A simulated transformation curve showing the induction time and the halfcompletion time........................................................................... 26 26 An ideal Master Curve for a transformation which obeys Cahn's relation for nucleation on two dimensional nucleation sites............................ 31 31 An illustration of diffraction by two scattering centers and the definition of the scattering angle 20 ............................................................................. 38 32 The scattering form factors for ideal spheres having radii: 80, 100, and 120 A .......................................................................................... ......... 42 33 A plot showing the effect of polydispersity on the scattering form factor for a gaussian distribution of spherical particles having a width parameter, = 1, 10, and 20 A ........................................................................................ 44 34 The modifications to the scattered profile which result from spheres having imperfect boundaries................................................................. 46 35 Model scattering profiles are shown for systems of spheres interacting via the hard core potential at varying particle concentrations, v4/v0 ..........................49 36 An illustration of Bragg scattering from a crystal lattice ................................. 51 51 A schematic of the data acquisition system............................................... 80 52 A top view of the polymer xray scattering furnace illustrating the relative placement of its various features as described in the text ................. 83 53 The final sample mount design is illustrated in a fron and top view ............. 87 54 A typical quench profile illustrating the base and sample temperatures as a function of time following the quench ........................................................... 90 55 A schematic of the SAXS collimation system............................................ 95 56 The chemical formulas of polystyrene and polybutadiene along with the selective solvent used in these studies, ntetradecane............................. 101 57 A graph of the beam height (out of scattering plane ) and width (in plane) profiles ................................................................................. 107 58 A graph of the measured parasitic scattering................................................. 110 61 Raw scattering profiles for the SB15 sample upon cooling.......................... 118 62 Raw scattering profiles for the SB15 sample upon heating.......................... 119 63 Raw scattering profiles for the SB25 sample upon cooling.......................... 120 64 Raw scattering profiles for the SB25 sample upon heating.......................... 121 65 Raw scattering profiles for the SB35 sample upon cooling.......................... 122 66 Raw scattering profiles for the SB35 sample upon heating.......................... 123 67 Raw scattering profiles for the SB50 sample upon cooling........................ 124 68 Raw scattering profiles for the SB50 sample upon heating........................... 125 69 Raw scattering profiles for the SBS25 sample upon cooling ......................... 126 610 Raw scattering profiles for the SBS25 sample upon heating........................ 127 611 Raw scattering profiles for the SBS35 sample upon cooling ....................... 128 612 Raw scattering profiles for the SBS35 sample upon heating........................ 129 613 A plot of the scattered profile for the SB50 sample at 44.0 C illustrating the estimated spherical form factor............................................ 135 614 A plot showing the three components of the parameterization of the low Q range of the scattering spectra..................................................... 138 615 A comparison of the simple cubic and bodycentered cubic structure specific fits to the scattered spectrum for the SB35 sample at 44.0 C......... 144 616 An illustration of the splitting of the firstorder Bragg reflection in the SB15 solution at 62.5 C................................................................ 150 617 Plots of the microdomains radius and lattice constants versus polymer concentration at approximately 44 C ......................................... 153 618 Graphs of the normalized peak intensities of the primary Bragg peak vs. temperature for each of the four polymer solutions..................................... 155 619 Graphs of the measured Bragg widths vs. temperature for the four polym er solutions ......................................................................................... 158 620 Graphs of the measured lattice constants vs. temperature for the four polymer solutions ................................................................................... 160 621 Variation of the amorphous peak amplitude vs. temperature for the SB15 solution ............................................................................................... 162 622 Plots of the amorphous peak position and width vs. temperature for the SB50 solution ......................................................................................... 164 623 The measured phase diagram for the polystyrenepolybutadiene /C14 solutions ................................................................................................. 166 71 A plot of the final spectrum as a function of quench depth for the SB15 solution ............................................................................................... 170 72 A plot of the final spectrum as a function of quench depth for the SB25 solution ............................................................................................... 171 73 A plot of the final spectrum as a function of quench depth for the SB35 solution ............................................................................................... 172 74 A plot of the final spectrum as a function of quench depth for the SB50 solution ............................................................................................... 169 75 A plot of the raw SAXS spectra as a function of time following a quench from 159.6 C to 112.2 C on the SB25 solution............................ 175 76 A plot of the raw SAXS spectra as a function of time following a quench from 130.6 C to 68.5 C on the SB15 solution............................ 177 77 A plot of the raw SAXS spectra as a function of time following a quench from 159.6 C to 94.0 C on the SB25 solution............................ 178 78 A plot of the raw SAXS spectra as a function of time following a quench from 170.0 C to 86.5 C on the SB35 solution............................ 179 79 A plot of the raw SAXS spectra as a function of time following a quench from 201.6 C to 98.5 C on the SB50 solution............................ 180 710 Graphs of the peak maximum as a function of time following a shallow quench ( upper graph ) and a quench to below the ordering temperature ( lower graph )........................................................... 182 711 A 3D plot of the temporal peak development of the scattered profile........... 183 712 Graphs of the fitted amorphous peak position and FWHM along with the Bragg FWHM and the measured lattice constant vs. time following a quench from 159.6 C to 94.0 C on the SB25 solution............................................. .................................................... 185 713 A plot of the normalized Bragg and amorphous fitted amplitudes as a function of time following a quench from 159.6 OC to 94.0 C on the SB25 solution.................................................................. 188 714 A plot of the normalized Bragg fitted amplitudes as a function of time following quenches from 160 oC to 102.2 OC, 94.0 oC, 82.0 oC, and 58.7 oC on the SB35 solution ................................................. 192 715 A plot of the quantity: In ( In ( ( 1 C )1 ) vs. the natural log of the time following quenches from 160 C to 102.2 OC, 94.0 C, 82.0 C, and 58.7 C on the SB35 solution...................................... 194 716 A master curve of all of the quench data on all of the samples as described in the text.................................................................................. 196 717 The time constants, t, for the ordering transition for the SB15, SB25, SB35, and SB50 solutions are plotted as a function of quench depth................................................................................................. 197 718 A schematic illustrating the proposed origin of the observed crossover in ordering behavior.......................................................................... 199 719 Graphs of the fitted and normalized Bragg amplitudes as a function of time following quenches from ~ 202 C to 152.0 oC, 143.5 OC, and 137.5 OC on the SB50 solution ............................................................. 201 720 Graphs of the raw data from a quench from 201.8 C to 152.0 C on the SB50 solution at various times following the quench.......................... 202 721 Plots of the fitted Bragg widths and positions along with the normalized intensity, Z, as a function of time following the quench from 201.8 OC to 152.0 C on the SB50 solution............................. 204 Bl A schematic of the heating circuit........................................................... 228 C1 A schematic of the solenoid interface............................................................ 230 D1 An illustration of the initial sample mount design....................................... 232 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN XRAY SCATTERING STUDY OF ORDERING IN BLOCK COPOLYMERS By Curtis Ray Harkless December 1990 Chairman: Stephen E. Nagler Major Department: Physics The block copolymer, a novel system for studying the kinetics of firstorder phase transitions, is investigated. Solutions of the block copolymer polystyrenepolybutadiene exhibit two types of phase transitions presently of great interest to the science community. Studies of the process by which these transformations occur can broaden our understanding of kinetic phenomena and aid in the identification of universal features such as nonequilibrium scaling. This thesis represents the first attempt to probe the kinetics of these transitions using synchrotron xray diffraction. The block copolymer molecule is composed of two different polymer chains joined by a covalent bond. When the chains are incompatible mesophases form through the process of microphase separation. The system also exhibits an ordering transition which results in a characteristic superlattice of the microdomains. A brief discussion of firstorder phase transition kinetics is given followed by a detailed review of the relevant literature on block copolymers. High quality diblock and triblock copolymer solutions were prepared. The structure of each system was determined from the xray scattering profiles as a function of temperature after which kinetic measurements were performed. Each kinetic measurement involved annealing the sample above the dissolution temperature and rapidly quenching the sample temperature to a fixed point below. The subsequent transformation process was observed through the xray scattering profile. Due to the resolution obtained at the synchrotron, the scattering contributions from the ordered and disordered states are identified and separated for the first time. As a result several new features are observed such as the presence of fine structure in the xray scattering profile. Fast kinetic measurements reveal that transformation occurs as a twostage process and that the ordering transition exhibits an unexpected crossover in behavior consistent with two dimensional nucleation. In addition, the velocities of transformation, as determined from the kinetic data, follow trends expected from fundamental thermodynamic considerations. Finally, novel oscillatory growth of the Bragg component is observed in the shallow quench limit. CHAPTER 1 INTRODUCTION One of the most fascinating and active fields in statistical physics is the study of phase transition kinetics.1 Specifically, there is much interest in understanding the non linear and cooperative phenomena involved in firstorder phase transitions (FOPTs).2 Although instances of FOPTs in nature are abundant, much of the essential physics of these processes remains unclear. Examples of these phenomena include phase separation in binary alloys,3 45, 6 and binary fluids, ordering in alloys,7. 8, 9.10.11,12.13 ferromagnetic and antiferromagnetic systems,14,15 and melting/crystallization transitions.16 One of the foremost efforts at this time is the identification of relevant parameters to form a universal classification scheme for these systems.17 Consequently, there is a need to study more diverse systems and polymer systems are and ideal and novel system for this purpose.1 A polymer is a long chain molecule composed of a number of repeat units called monomers. Generally, each monomer itself is an organic molecule joined to adjacent molecules by a covalent bond. Common examples of polymers include polystyrene, polyisoprene, and poly (vinyl chloride). The excellent mechanical properties of these materials lend themselves to a variety of industrial applications. The interesting statistical physics involved in understanding these materials has attracted the attention of many notable physicists including de Gennes, Debye,18 Kramers,19 and Flory. Excellent references on the static and dynamic properties of polymers have been written by Doi and Edwards,20 Flory,21 and de Gennes.22 The block copolymer molecule is formed by joining two chemically distinct polymer chains end to end with a covalent bond and is the subject of the experimental studies to be presented here. A typical block copolymer is illustrated in Figure 11. Much recent work has been done toward understanding these complicated systems.23 When the monomers, A and B, are incompatible ( AA BB, nearest neighbor configurations are favored over AB configurations ), the system attempts to minimize internal energy by forming A or Brich domains thereby reducing the surface to volume ratio between A and B regions. This type of arrangement leads to a loss of conformational entropy, resulting in an equilibrium structure that is a balance between these two opposing forces. At high temperatures, the entropic force dominates and the system is a homogeneous melt as shown in Figure 11. As the temperature is lowered, the balance shifts to formation of domains at the dissolution temperature, Td. This domain formation is similar to the phase separation observed in binary alloys. However, the junction of A and B monomers in the block copolymer limits the ultimate size of the phase separated regions to microscopic dimensions. This process is appropriately termed microphase separation and results in the different domain morphologies observed in these materials.24 Phase separation in binary systems has been well studied and has led to the formulation of very general relations describing the phase transition kinetics in conserved order parameter systems.' Microphase separation is an intriguing analog of this process involving both the complex process of polymer interdiffusion and an inherent constraint to the size of the phase separated regions. For this reason, block copolymers are useful systems for broadening our knowledge of kinetic phenomena and testing the limitations of present models. A variety of domain morphologies has been observed in block copolymer systems through xray scattering23 and transmission electron microscopy2.26 and the type of morphology was found to depend on the fractional composition of the individual block THE BLOCK COPOLYMER MOLECULE AAAAAAAABBBBBBBB MICROPHASE SEPARATION T > Td T < Td Figure 11 The diblock copolymer molecule and an illustration of microphase separation. copolymer chains. For example, polymers having a small A block relative to the B block are known to form spherical Arich microdomains in a Brich matrix. Remarkably, these systems are seen to exhibit longrange order in the form of a cubic lattice of the spherical microdomains. The specific type of cubic structure is presently the subject of some debate27 and, for this reason, structure determination is discussed in detail in this work. In those copolymers where the A block is somewhat smaller than the B block, a 2dimensional hexagonal closed packed ( HCP ) arrangement of cylindrical domains is formed. A lamellar structure results if the A and B blocks are of comparable size. These features are summarized in Figure 12. Often in studying these materials, it is useful to employ a solvent which selectively dissolves one block of the copolymer. In this way the interaction potential between unlike monomers is partially screened resulting in a lower, experimentally accessible dissolution temperature. An added benefit of using selective solvents is the ability to control the effective fractional composition of the chain by selectively swelling one block relative to the other. The study of these polymer solutions is also important from an industrial standpoint in that additives are commonly incorporated in pure polymer materials to improve their already advantageous mechanical properties. Further interesting effects were observed in small angle xray scattering (SAXS) studies of polystyrene(PS)polybutadiene (PB) diblock copolymer in the selective solvent ntetradecane (C14).28 The system forms spherical PS domains within a PB/C14 matrix. However, a temperature range was found below the dissolution point where the expected higher order Bragg reflections were missing from the SAXS profile indicating the absence of longrange order. At lower temperatures, evidence of higher order Bragg reflections was found leading the authors to propose an ordering temperature, T,, below which the spherical microdomains arrange themselves to form a cubic lattice. This ordering process is illustrated in Figure 13. 4c CD C) 0 < m U) ! MA II (I) II C) CD H II The three domain morphologies commonly observed in diblock copolymer systems. 0 C< F > :z I CO U) D m U) Figure 12 0 7 CD 0 U 0 IF 0 < IF 7> 0 I 0 C) rH o rT CO A DISORDERED ARRANGEMENT OF SPHERES B / Cli ' 4 5 / cli B / C14 B / 1 5 / Cli AN ORDERED LATTICE OF SPHERES Figure 13 An illustration of the ordering of spherical microdomains onto a cubic lattice. Microdomain ordering in block copolymers is an interesting analog to melting crystallization phenomena and as such is characterized by a nonconserved order parameter (NCOP). Other systems such as ordering in "polyballs"29 have contributed much to our understanding of these phenomena. The presence of this second type of transformation in block copolymer solutions provides an additional opportunity to test the applicability of present models of crystallization and behavior in NCOP systems. An important feature of block copolymer materials is their great diversity in industrial applications. These include their use as synthetic rubber, additives to plastics and adhesives, and hydrophilic coatings.30 Much theoretical and experimental research has been done in attempting to relate the mechanical properties of block copolymers to their structural properties.31 32 For this reason, a better understanding of the structural and thermal behavior of these materials is of direct industrial significance. The SAXS technique is an excellent method for analyzing the structure of block copolymers because the scattered intensity is directly related to the positional correlations of the particles in the system. It was the goal of these studies to perform high resolution xray scattering measurements of the equilibrium structure of block copolymer solutions and to probe the microphase separation and ordering processes with fast kinetic measurements. The successful completion of this experiment was dependent on three major requirements. These were the use of high quality samples, a specially designed polymer xray scattering furnace that allowed for rapid insitu quenches, and a very intense xray source enabling high resolution measurements. Ultimately, xrays produced by synchrotron radiation were utilized. The experimental methods employed and the novel results obtained are discussed in the following chapters. This dissertation is divided into seven chapters. Chapter 2 is a summary of recent advances in our understanding of first order phase transitions and provides a background for understanding the significance of these measurements. The third chapter is an in depth discussion of the general principles of xray diffraction and relates the relevant 8 physical parameters to experimentally measured quantities. Chapter 4 is a brief review of the relevant literature on block copolymers. Chapter 5 is a detailed description of the experimental apparatus and procedures performed in these studies. The parameterization, analysis, and interpretation of the equilibrium results are presented in Chapter 6 in preparation for discussion of the kinetic results in Chapter 7. Finally, the important conclusions which are drawn from this work will be summarized in Chapter 8. CHAPTER 2 PHASE TRANSITION KINETICS This chapter is a brief review of the major developments in firstorder phase transition kinetics. The first section provides the definitions of many of the basic concepts as well as an introduction to some of the general behavior. Broad comparisons are made with higher order critical phenomena. The transformation process is divided into three temporal regimes which are then discussed individually. These are the early, growth, and coarsening regimes. Two distinct mechanisms are identified in the early regime: nucleation and spinodal decomposition. During the growth regime, transformation is characterized by growth laws and transformation curves. Finally, the late stage of growth, coarsening, is described. Phase Transition Kinetics General Concepts Figure 21 is an example of the phase diagram for a binary alloy exhibiting a phase transformation. The solid line on the plot represents the transition temperature as a function of sample composition. The dashed curve is called the spinodal curve and will be discussed shortly. The objective of kinetic studies is to observe the process by which the transformation from one phase to another occurs. One method of probing these phenomena is through the thermal quench. In a typical quench experiment, the system is held at a temperature, Ti ( or other extensive variable ), above the transition temperature, T, and then rapidly lowered to a fixed value below T,, Tf. If the quench occurs in a time short relative to the time required for transformation, the system will still have the LU _ T DISORDERED COEXISTENCE r0 [ CURVE Ln SPINODAL  i \ CURVE LU I __ UNSTABLE SMETASTABLE COMPOSITION Figure 21 An example of a phase diagram for a binary alloy exhibiting a phase transition. structure of the initial phase. This phase is now unstable ( or metastable ) and will decay to its equilibrium phase as time proceeds. The processes by which transformation occur are often complex and the study of these processes is one of the most challenging topics in nonequilibrium statistical physics.' In order to quantify the transition process a quantity called the order parameter, 4, is defined. This parameter takes on values between zero and one, being identically zero in the disordered phase and one in the fully ordered phase. In some systems the order parameter is a conserved quantity and in these cases, 4 changes only locally and remains on average a constant for the entire system. Systems of this type are called conserved order parameter ( COP) systems. An example of a COP system is a binary alloy which exhibits phase separation.3 4, 5,6 A typical nonconserved order parameter ( NCOP ) system is a binary alloy which exhibits an orderdisorder transition.7, 8.9. 1.12.13 An example of such a system is Cu3Au.7, 10 In the disordered state the crystal structure for this system is facecentered cubic (FCC ) where the Cu and Au atoms are randomly distributed on all sites. In the ordered state, the structure remains FCC with the Au atoms dominating the conventional cube covers and the Cu atoms occupying the cube faces. The order parameter is related to fraction of each species on a sublattice. Since the order parameter in this case is not constrained by conservation laws, this is an example of an NCOP system. Other examples of transitions characterized by NCOPs are melting/crystallization phenomena16 and antiferromagnetic ordering.14,15 The driving force for transformation is the difference in free energies between the initial and final phases. Phase transitions may be classified as first, second, or higher order based on a fundamental thermodynamic criterion according to the Ehrenfest classification scheme.33 If the first derivative of the free energy of the system with respect to the order parameter is singular, the transition is said to be first order. One characteristic of first order phase transitions ( FOPTs ) which follows directly from the discontinuity in the free energy is the existence of a latent heat (i. e. the heats of melting and vaporization ). In addition transitions involving a change of symmetry from one symmetry group to another that is not a subgroup of the first can be identified as FOPTs because it is not possible for the system to go from one phase to the other continuously. Discontinuity in the second and higher derivatives of the free energy with respect to the order parameter without singularity in the lower derivatives implies second and higher order transitions. Second and higher order ( also called critical or multicritical) phenomena are in general much better understood than first order phenomena.1.2 This is because critical phenomena arise from fluctuations in the order parameter which can be characterized by a single correlation length. In fact, near the critical point the correlation length diverges and only phenomena occurring on this length scale are relevant to the transition behavior. It is the existence of this dominant length scale which has led to the success of the renormalization group approach and the observation of scaling and self similarity in these systems.34 First order phase transitions are different because there is in general no dominant length scale and transformation is determined by processes at the interface between transformed and untransformed regions. Mean field theories express the free energy, F, as a functional of the order parameter, however, approximations are required in evaluating this expression. The equilibrium state of the system can be determined by minimizing the free energy with respect to <. The GinzburgLandau35 formalism describes an expansion of the coarse grained free energy in powers of the order parameter. In general, the order parameter may be a scalar, vector or a tensor as is the orientational ordering of liquid crystals.2 The prototypical shape of the free energy curve for a scalar order parameter and fixed composition is shown in Figure 22 for several temperature ranges. F{j) has the double well structure characteristic of phase transformations. At temperatures above Te, the lowest energy state corresponds to <=0. As the temperature is lowered, the difference in free energies between the ordered, 4=1 and disordered, 4=0 configurations is reduced T > Tc T Tc Tc > T > Ts Ts > T The shape of the free energy curve versus order parameter for various temperatures, above and below the transition point Figure 22 until T=T(, at which point the free energy is equivalent for both phases. Below T, the ordered phase is favored, but the state 0=0 is at a local minimum of free energy and is therefore metastable. If the temperature is lowered further, the potential barrier separating the ordered and disordered phases disappears and the <=0 state becomes unstable. This instability limit which separates the metastable and unstable regions of the phase diagram is commonly called the spinodal curve.'.2. 35 In fact it is no longer believed that a sharp spinodal point exists2 as will be discussed momentarily. The process by which the disordered state decays is qualitatively different for quenches into the metastable and unstable regions of the phase diagram. In fact two distinct behaviors are associated with these two regimes. When the initial phase is metastable, transformation is said to result from the mechanism of nucleation. During nucleation, localized droplets of the ordered phase spontaneously form within a disordered matrix. These droplets grow independently until separate domains impinge on one another. When the initial phase is unstable, it will decay through the mechanism of spinodal decomposition. During spinodal decomposition long wavelength, infinitesimal oscillations of the order parameter appear which are characterized by an amplitude and wavelength that grow with time. Many of the essential features observed in phase transition kinetics have been elucidated through analysis of very simple models.' Perhaps the most useful of these is the Ising Model because of the ease of calculation and the straightforward analogies to physical systems.16 Within the Ising model, a lattice is composed of spins having values of o = + 1. By choosing the correct spin dynamics and an appropriate order parameter this model may be applied to both NCOP and COP processes. For example, if changes in the lattice distribution of spins occur by exchanging adjacent spins the system is said to obey Kawasaki spin dynamics.' The total magnetization M=Yoa (21) i remains constant throughout the transformation and is therefore analogous to the order parameter in a COP system such as phase separation in binary alloys.36 In this case "up" spins correspond to one type of atom while "down" spins correspond to the other type of atom. The orderdisorder transition in binary alloys can also be modeled using Kawasaki dynamics if the order parameter is defined to be a sub=lattice magnetization: M,= Doi (22) sublattice In this event, the total magnetization is conserved, but the order parameter is nonconserved.37 38 Another way to model NCOP systems is with Glauber spin dynamics. Within this model spins are able to change sign independent of adjacent spins so that the order parameter which is taken to be the total magnetization is nonconserved. In general spin dynamical models have shown good agreement with theory.16 Conceptually the transformation process can be divided into three temporal regimes: early stage growth characterized by either nucleation or spinodal decomposition, intermediate growth, and late stage growth commonly called coarsening. These three stages are illustrated in Figure 23. During late stage growth, essentially all of the system is in small fully transformed regions. During this regime the larger of these domains will grow at the expense of the smaller domains resulting in an increase in the average size of the domains. This process is termed coarsening and is illustrated in frames c and d of Figure 23. The different time regimes and transformation mechanisms will now be discussed separately beginning with nucleation and spinodal decomposition, followed by a brief discussion of intermediate stage growth and finishing with a description of coarsening behavior. The chapter will conclude with some comments on recent efforts to develop a universal classification scheme. Nucleation Growth 0 Growth 0* Early Coarsening Late Coarsening An illustration of the different temporal regimes in the transformation process. Figure 23 Nucleation Theory The rate of transformation from the old to new phase is controlled by the combination of the total nucleation rate and the growth rate. The total nucleation rate per unit time, IT(t), is the product of the nucleation rate, I(t), and the volume available for nucleation, V,. Conceptually, the behavior of the total nucleation rate can be divided into three regimes. Often these three regimes are distinct but in some cases or there may be significant overlap. In addition, not all of the regimes are necessarily observed. The first regime is characterized by a transient nucleation rate.39 Following a quench from above the transition point I(t) will be zero initially and then grow smoothly to its steady state value. Once the nucleation rate reaches its steadystate value, Is', it will remain unchanged throughout the transformation. However, as the transformation proceeds the nucleation volume, V,, decreases leading to a decrease in the total nucleation rate, IT(t), in the late time regime. The process by which the nucleation volume is exhausted depends in detail on the type of site in which nucleation occurs.40 Several of these processes will be discussed in a later section. Each of these three regimes will be discussed in detail in the following sections. The driving force for transformation from the metastable state to the equilibrium state is the free energy difference between the two phases.39 However, when a finite sized nucleus forms, there exists an interfacial region between the nucleus and the bulk metastable phase. The surface tension serves to raise the free energy of the droplet. The total free energy of the embryo is typically written as the sum of bulk and surface terms: AF, = n (f f) + So (23) or AF,= [ (f f f ) + 4xcr2 (24) In the above expression, the droplet of radius, r, is composed of n particles. The product of the surface area of the droplet, S,, and the surface tension, a, gives the surface contribution to the free energy. The bulk free energies of the equilibrium and metastable phases per particle are f" and f", respectively. The volume per particle in the equilibrium phase is designated v". The volume free energy term f" is generally assumed to be independent of droplet size. Similarly, a is assumed to be independent of size and calculations of the surface tension exhibited by crystalline clusters suspended in liquid support the validity of this assumption.41 The energies, fe and f", are in general a function of the reduced temperature, t=(ToT)/To where To is the transition temperature. Above To, fe > fm and AFn is a monotonically increasing function of n and fluctuations of any size tend to evaporate. In this case, the equilibrium form of the distribution of sizes of nuclei as determined by Boltzmann statistics is given by fAFn1 N. = N' exp (25) which is only of significant magnitude at very low values of n. However, below To, f < fn and AF, goes through a maximum at the critical radius, r,. 2ov* rc =(ife) (26) Classically, nuclei of this radius are in unstable equilibrium whereas nuclei with r < r, will evaporate and droplets with r > r, will grow. This model, while being somewhat simplified, exhibits the important feature that nuclei having radii near r, control the nucleation rate. In other words, it is the rate at which fluctuations reach the critical size that determines the rate of decay of the metastable state. Immediately following a quench the majority of nuclei are of a very small size and until the first droplets reach the critical size the nucleation rate is effectively zero. This period is called the induction or incubation time, to. There are two commonly used approximations for I(t) in the transient period. The first assumes an incubation time followed by a slowly growing nucleation rate which is approximated by a power law, I(t) = ( t to ) B,t (27) Here the step function, O(t), incorporates the effect of a finite incubation time. This approximation will be most valid when the nucleation rate is a slowly changing function of the time and the transformation nears completion before the nucleation rate approaches its steadystate value. The second approximation incorporates an induction time and a step function to describe I(t) and will be most accurate when, following the induction period, I(t) grows rapidly to its steadystate value. Typically, the effect of the changing nucleation rate is most obvious in the early stages of the transformation. Experimental measurements are not overly sensitive to the exact form chosen to approximate the time dependence of I(t) in this regime.38 In the second time regime the nucleation rate is constant at its steadystate value, I". While there have been many attempts to calculate I" by far the most successful treatment is that of Becker and Doring.42 The key to this model is its treatment of the problem as a kinetic phenomenon. A set of difference equations is used to describe the number of droplets as a function of size and time, Z,. The rate of change of the number of droplets composed of n particles is expressed as the difference between the rate of condensation and the rate of evaporation of particles from the nucleus. The condensation rate is assumed to be proportional to the surface area of the nucleus, S., and otherwise independent of n. As time proceeds, the number of embryos of size n approaches its quasisteady state value ( Z., = Z ). This implies that I, = I, a constant independent of n so that the distribution of nuclei, Z7, is constant. There are two boundary conditions for the kinetic process. The first is that for small n the distribution of nuclei approaches its equilibrium form as determined by Boltzmann statistics (Z, approaches N, from Equation 25 ). The other boundary condition is that embryos containing a very large number of particles leave the system completely ( Z,= 0 n>>n,). By equating the rates of production and loss of embryos of size n, the following result is obtained: Is = v (28) n=l where v is the rate of condensation per unit surface area of the nucleus. Since AF exhibits a maximum at nc, the quantity N,1 from Equation 25 will be sharply peaked near nc. Becker and Doring reasoned that only values of n near the critical number would contribute significantly to the above sum. Taking the continuum limit of N, and AF, the standard BeckerDoring form for the steadystate nucleation rate is obtained: I = A exp (29) fNvSc] AF(n.) where A= 3J B = AF(n) (210) In the above relations, both the prefactor A and the energy term, B, depend on the reduced temperature. In those cases where these parameters are only weakly dependent on the temperature, the steadystate value of the nucleation rate takes the Arhenneus35 form with an activation energy equal to the free energy difference between a critically sized nucleus and the bulk metastable phase. Spinodal Decomposition The different modes of transformation observed for metastable and unstable systems can be viewed as instabilities against different types of fluctuations as discussed by many authors.1' 2.16 The method of decay of a metastable state is through the nucleation process. This indicates an instability against localized, large amplitude fluctuations of the order parameter, the size of which grows with time. This process is illustrated in Figure 24. In contrast, the method of decay of an unstable system is through a process termed spinodal decomposition. This process is also shown in Figure 24. This mode of decay indicates an instablility to nonlocalized or long wavelength fluctuations of the order parameter of very small amplitude. The amplitude of these fluctuations grows with time. While nucleation gives rise to growing droplets, spinodal decomposition leads to complex interconnected patterns of the transformed phase.43 The interfacial region between transformed and untransformed regions is initially much less defined than that of nucleated droplets but sharpens as time proceeds. As the size of the transformed domains increases, the patterns become less interconnected until the domains are similar in shape to those observed for nucleated transformations. The significance of the spinodal curve has recently been questioned by many researchers2 17 because the concept of an instability limit is an artifact of mean field theory used to formulate the free energy functional. However, real systems do show the types of behavior described above. And although the transition from nucleated to spinodal decomposition behavior occurs gradually26 instead of suddenly at the spinodal point, the concept of a spinodal limit is not totally without merit. NUCLEATION ti \ t2 t t3 Distance SPINODAL DECOMPOSITION Distance A contrast of the nucleation and spinodal decomposition transformation mechanisms. Figure 24 Domain Growth For the types of systems we are considering, the growth of domains of the equilibrium phase is said to be thermally activated. This implies that the growth velocity is a function of temperature, being greatly reduced at temperatures low relative to the transition temperature. One consequence of this behavior is the ability to "quench in" the high temperature phase by rapidly lowering the system temperature to a value where the velocity of growth is extremely small. In addition to the dependence on temperature, the growth velocity, F, will also have a characteristic dependence on the size of the domain ( or equivalently, on the time elapsed since the birth of the domain). In general, thermally activated transitions can be divided into two groups: interface controlled and diffusion controlled reactions.39 These two groups are characterized by functionally distinct forms for the growth velocity. Generally, transformations between two phases having approximately the same particle concentrations are controlled by the energetic of processes occurring at the interface. This being the case, the growth velocity, F, is independent of particle size and time so that the average dimension of a growing droplet, L(t) satisfies the simple equation.44,45 L(t) = ro t, F(t) = r (211) Christian39 has estimated the dependence of F on the temperature, T, for the decay of metastable phases. This calculation assumes the difference in free energy between the old and new phases to be small relative to the thermal energy, kBT, where kB is the Boltzmann constant. In addition, the characteristic frequency for particle motion, vo, is assumed to be identical in both the transformed and untransformed regions. The resulting relation is (T) = ( AFv) e (212) r(T) =(vo) kgT exp [ T(212) ^B1 I B1 Here 8 is the thickness of the interface and AH, is the activation energy separating the equilibrium and metastable states which is generally a function of the reduced temperature, T. In the event that there is a significant concentration difference between the initial and final phases, it is possible for the propagation of the domain boundary to be diffusion limited. If the concentration of particles is higher in the transformed domain, there exists a region adjacent to the interface called the depletion zone' in which the particle concentration falls below its average value in the untransformed volume. In order for the boundary to propagate, particles must diffuse through the depletion zone and then "condense" on the transformed domain. If the processes occurring at the interface are much slower than the diffusion of particles through the depleted region, the transformation is interface controlled (i. e.. L(t) = F0 t). However, if the condensation process is rapid, transformation will be limited to the rate of diffusion through the depletion zone. From the diffusion equation and dimensional analysis Christian finds D L(t) ( Dt )z r(t) (213) where D is the diffusion constant. This parabolic growth law is verified in more exact calculations,46.47 and in computer simulations.48'49 It is noted that the above growth laws for interface and diffusion controlled growth are only valid prior to impingement of one transformed domain on another. Different growth laws govern behavior in the postimpingement or coarsening regime and these will be discussed shortly. Transformation Curves Now that the processes of nucleation and growth have been discussed, a method for combining the results to yield a rate of transformation will be presented. A measure of the degree to which transformation has progressed is the fractional volume in the transformed state, C(t). From the previous discussion of the nucleation rate we expect this quantity to grow slowly at first and then more rapidly as a larger number of embryos reach the critical size and begin to grow. Eventually, the surfaces of different domains will impinge on each other, slowing the growth of ((t) at later times. The shape of a typical transformation curve is shown in Figure 25. The relevant features include the induction time, to, and the time to halfcompletion, ;,. The earliest formal theories of transformation kinetics are due to Kolmogorov,50 Avrami,5152.53 and Mehl, and Johnson.54 The argument of Avrami goes as follows. If a nucleus forms at some time, t, then the transformed volume which is contained in the nucleus is given by V(t) =rlyxyY (tt)3 (t>t) (214) =0 (t Here Yx, y,, and yz are the growth velocities in the x, y, and z directions respectively and 4L Tr is a shape factor (for isotropic growth, rl = and y, = yy = y, = y). The above expression is valid for a system exhibiting a linear growth law, L(t) t, such as is the case for an interface controlled transformation as was previously discussed. In a system where growth is diffusion limited, the growth law is given by L(t) tIA. In this case Equations 214 would be replaced by V = TlyYy z ( t t)3' (t>r) (215) =0 (t For the remainder of this section only systems exhibiting a linear growth law will be discussed with the understanding that the appropriate results for different growth laws can be derived simply by making the modification shown in Equations 215. _ _ 1/2 / 0 TIME A simulated transformation curve showing the induction time and the halfcompletion time. S C. Figure 25 i 1 I I In early times there is little impingement of domains, so that each droplet grows independently. In this limit the transformed volume is given by V= 3]f I(t) (t T)3 d' (216) 0 where isotropic growth was assumed. For a constant nucleation rate, I(t) (I(t) reaches the steadystate value, I", immediately ) Equation 216 is reduced to = V 3 I3 t4 (217) This confirms the rapid increase of C at early times. Unimpeded growth will not continue for all times, however, due to impingement by other droplets. To deal with this complication, a concept called the extended volume, Ve is introduced. The extended volume is the volume of transformed material assuming that nuclei continue to form not only in untransformed regions of the system, but in previously transformed regions as well. Similarly, growth of domains is assumed not to stop when impingement occurs; rather, each domain continues to grow through the others. In this view, Ve is equivalent to VB in Equation 217 for all times. In any increment of time, dt, the probability that the incremental increase in the extended volume occurs in a previously untransformed region is ( 1 VP/V ). It follows that dVP= (1 VB/V) dVe (218) or = vP/V = 1 exp( Ve) = 1 exp(T_ I" t4) The shape of this curve is shown in Figure 25 and in the early time this expression reduces to Equation 217. In the above argument, the nucleation rate I(t) was assumed to be a constant. In general, the total nucleation rate is a function of the time, either due to transient nucleation in the early regime or due to nucleation site saturation in the later regime. The corresponding transformation laws may be derived in a straight forward way by inserting the appropriate expression for I(t) into Equation 216. These calculations generally lead to an expression similar to that in Equation 218 except that the time exponent is different from 4. This led Mehl, Johnson, and Avrami ( MJA) to propose the following general transformation law39: = 1 exp( ktn) (219) For a constant nucleation rate it has been shown that n=4. For an increasing nucleation rate ( early regime ) n>4, and for a decreasing nucleation rate ( site saturation) n<4. For homogeneous nucleation, often the nucleation rate may be approximated by a step function being zero during the induction period, to, and then taking on its steady state value Is. In this case, the transformation curve is given by Equation 219 with n=4 and the time, t, is replaced by tto. Often nucleation occurs not homogeneously, but at impurity or defect sites within the sample. The transformation curves for heterogeneous nucleation can be calculated in much the same way as for the homogeneous case if the appropriate form for the nucleation rate can be found. For instance, when nucleation occurs spontaneously at No impurity sites the effective nucleation rate, I,(t), may be represented by a delta function at t=O having a strength, No; i. e.. I(t)= No 8(t) (220a) so that from Equation 217, = 1 exp( N 'y t3) (220b) The form of the transformation curve is unchanged, with n=3 reflecting the nature of the nucleation process. Interesting dimensional effects result if nucleation is spontaneous and growth is restricted to either a lamellar or rodlike volume. For growth occurring in a sheet ( 2D growth ), as might be the case in a thin film, the extended volume is given by : VP = (47cV6)) No y2 ( t x )2 (221) where 8 is the thickness of the sheet. The result for a constant nucleation rate is the MJA expression with n=2. Similarly, it is seen that for growth in a rodlike volume ( 1D ) the result is MJA with n=l. More complicated transformation curves may result if nucleation occurs heterogeneously, but not spontaneously. If defects are located randomly, the growth in early times should be indistinguishable from the homogeneous case ( Equation 219 with n=4). In time the available nucleation sites may be exhausted before transformation of the entire volume is complete. In this event, different behavior is expected following saturation of the available sites. Cahn derived an expression for nucleation occurring on grain boundaries, edge dislocations and point defects.40 Analogous to the concept of an extended volume for transformation, an extended volume for nucleation is introduced. In this way, the space available for nucleation at time, t, is calculated as a function of the growth velocity, steadystate nucleation rate, and the initial volume available for nucleation, Vn. The resulting transformation curve for nucleation occurring on two dimensional defects is : S= 1 exp(b f(t)) (222) 1 where f(t) = ( 1 exp(' ( (132+2l3)]d 0 The coefficients a and b are related to the physical quantities y, I", and V, in the following way: a= ( Iss )1t3 (223) and b = 2Vn l The function f(t) has two limiting forms, at late times f(t) t, and at early times f(t) t as expected. The transition from n=4 before site saturation to n=l after saturation is best illustrated by a plot of In( ln( 1/(1) ) ) vs In( t) as shown in Fig. 26. In this plot the slope of the curve is equivalent to the exponent in the MJA form, and the bend in the line corresponds to site saturation. From Equations 221 it is evident that if we offset the plot vertically by In( b ) and horizontally by ln( a) we have a master curve for this type of transformation. In other words, plots for a number of data sets will fall upon this master curve if offset by the appropriate selection of a and b on this scale. Similarly, the expressions for the transformation curves for nucleation on line and point defects have been derived. In both cases the early time behavior agrees with the MJA form having n=4. In late times, n crosses over to 2 for nucleation on line, and 3 for point defects. It is apparent that the spontaneous nucleation of point defects discussed previously is just a special case of this result where only the n=3 regime of the The Cahn Model 8 S6 4 2 0 2 C 4 6  6  8 1 Master Curve 0 1 2 3 4 In ( Figure 26 time ) An ideal Master Curve for a transformation which obeys Cahn's relation for nucleation on two dimensional nucleation sites. n = 1 n = 4 transformation curve is observable. In fact, the easily observable range of C in experiment is typically from about 0.01 to 0.99. Although it seems that this range should encompass the full range of kinetic behavior, this range corresponds to only a relatively small portion of the master curve. If the nucleation rate is too high, saturation occurs very early in the transformation and only the behavior of saturated regime is evident. Conversely, if the nucleation rate is too low, saturation occurs late in the reaction and only the exponent corresponding to the unsaturated or homogeneous regime is seen. In fact, the crossover is only visible if saturation occurs when t 0.5. The steadystate nucleation rate within the Cahn theory for nucleation on 2D defects which corresponds to this point is given by : r i3 Is = V3 (224) Systems exhibiting the required balance between the nucleation rate, growth velocity, and nucleation volume appear to be exceedingly rare. As will be discussed, the observation of this phenomenon is one of the more fascinating features in the study of ordering in block copolymers. Late Stage Growth Coarsening Often the equilibrium state toward which the system is evolving is degenerate. For example, in the Cu3Au system described previously there are four identical sites in the unit cell. The lattice can be defined by choosing any one of these four sites as the corer site. In this way it is seen that the ground state degeneracy of the ordered phase is p=4. For the crystallization process, p=o reflecting the fact that the crystal may form in any orientation within the liquid.39 In the event the ground state is degenerate, in the late time regime the system will be composed of many fully transformed domains distributed in the p degenerate states. As time proceeds, the number of domains will decrease with a corresponding increase in the average size of the ordered domains. This process is called coarsening. Within the coarsening regime, the system is selfsimilar with respect to scaling by length and time. This can be illustrated by zooming in on frame (c) of Figure 23 and comparing the result with frame (d). The relevant features in each case will on average be identical. A well known consequence of this type of scaling is the existence of a characteristic length37, L(t), (the domain size) where L(t) t (225) Here t is the time following the quench. The exponent a has been seen to be different in COP and NCOP systems. Specifically, theoretical calculations55 and experimental observations56 of COP systems support the assertion a=l/3. In NCOP systems the growth rate of domains in the coarsening stage is determined by energetic at the interface and is often curvature driven. Theoretical calculations predict a corresponding growth exponent, a=l/2.57 This result is supported by a growing number of experimental findings7.58 and computer simulations59,60. It should be noted that the growth exponents are different in the coarsening regime from those discussed for the growth regime. This is because during the early stages, growth of domains is relatively uninhibited by impingement. In the coarsening regime, the domains are strongly interacting with the major mode of growth being the absorption of smaller domains by larger ones. Also, the dynamic quantity during coarsening is the average size of a domain, while the volume fraction of the system in the transformed phase remains approximately unchanged. In the early stages of transformation it was seen that impurities and defects have a profound effect on the nucleation process by providing heterogeneous nucleation sites. These entities can also lead to interesting effects in the coarsening regime. Lai et al.61 show that random fields which are generated by impurities and defects slow the coarsening process so that the growth law becomes: L(t) (log(t) )m (226) where m depends on the detailed nature of the imperfections in the system. Evidence supporting this assertion has been provided by experiments on Cu3+xAu where excess Cu atoms appear to behave as diffusive impurities.10 Universal Classification The nonequilibrium scaling behavior observed at late times is reminiscent of the scaling seen in critical phenomena. For this reason, one of the foremost efforts in the study of phase transition kinetics has been the attempt at developing a universal classification scheme analogous to that for critical phenomena.17 In this way it is hoped that the relevant parameters such as ground state degeneracy, dimensionality, and conservation property of the order parameter can be identified so that the behavior of widely diverse systems can be better understood. The coarsening growth law is one feature which may be useful in identifying these universality classes. For systems exhibiting random fields, the growth law is logarithmic suggesting that systems with different types of impurities might form separate universality classes. The growth exponent a has been seen to depend on the type of order parameter. It is, however, independent of ground state degeneracy, and dimensionality in many cases. In order to better distinguish the relevant parameters for classification and provide a deeper understanding of the fundamental processes which govern phase transformation, a broad range of physical systems must be investigated. Polymers and block copolymers in particular are excellent systems for this purpose.1 An excellent method for observing the transformation process in these materials is through the small 35 angle xray scattering spectrum. The fundamentals of the xray scattering technique are now presented. CHAPTER 3 THE XRAY SCATTERING TECHNIQUE The xray scattering technique is an excellent probe of inter and intraparticle correlations. Many excellent reviews of general scattering theory can be found in the literature.62.63.64,65.66 A brief discussion of the fundamentals of xray diffraction is given here followed by some details of intraparticle effects in small angle scattering. Interparticle interference effects resulting from a system of particles with liquidlike correlations are then described. Finally a description of the scattering from an ordered array is presented including the effects of grain size, strain, and temperature. Fundamentals of Xray Diffraction When a beam of xrays is incident on a sample, a characteristic scattering pattern results. In the systems with which we are dealing, the diffraction of xrays is due almost entirely to elastic scattering from the electrons in the material. The scattering distribution from a single electron is given by the Thomson formula:67 Ie(20) = I0 m2 (31) The angle 20 is called the scattering angle and is illustrated in Figure 31. Here OP is a polarization factor which equals 1 for an incident beam polarized perpendicular to the plane of scattering ( as is the case for synchrotron radiation). For an unpolarized beam, l+cos2(20) = 2 (32) In either case the polarization factor is approximately equal to unity in small angles. Compton and inelastic scattering are negligibly small at small angles (i.e. 20 < 20 ).64 The elastically scattered waves are coherent, meaning that the total scattered amplitude is the linear superposition of the electric field vectors for each scattered wave. The scattering from two scattering centers is illustrated in Figure 31. If the amplitudes of the two scattered waves are equal and of unit magnitude, the total scattering amplitude will be determined by the relative phases of the two waves. The phase for each wave at the point of detection equals 2nx/ times the optical path for each of the two waves. Actually, there is an additional factor of n which results from the phase inversion incurred by scattering from an electron. Since all scattered waves are offset by this factor, the total phase shift is given by 27t = r. ( so )=r.(kko ) (33) Here, so and s are unit vectors in the incident and scattered directions, and k k are wave vectors in the incident and scattered directions. At this point it is convenient to define the scattering vector, Q, which is the difference in incoming and outgoing wave vectors so that [20 4E f29 Q = kko, Q= 2k sin = sinm (34) The scattering vector is the variable typically used in xray diffraction and henceforth the scattering will be described in terms of this quantity. This is done with the understanding that Q can be related to the scattering angle, 20, by Equations 34. The transformation from Q to scattering angle is approximately linear at small Q with 10 = 0.07121 A1 for an incident wavelength of 1.541 A. With these definitions, the amplitude of scattering from two scattering centers is A = 1 + exp(i4) = 1 + exp( iQr) (35) So An illustration of diffraction by two scattering centers and the definition of the scattering angle, 20. Figure 31 7(sSo) This result can be extended to the case of scattering from a continuous system characterized by an electron density, p(r). The resulting amplitude is given by A(Q) = fdVp(r) exp( iQr) (36) Hence, the scattering amplitude is equivalent to the Fourier transformation of the electron density. The observed scattered intensity is related to the amplitude in the following way: I(Q) c AA*= f dV1 dVz p(r1) p(rz) exp(iQ(rlr2)) (37) Redefining r = r, r2, in Equation 37 yields the result that the scattered intensity is equal to the Fourier transform of the autocorrelation function of the electron density, 2 p (r): I(Q) f dV 2(r) exp( i Qr) (38) where p (r) = dV, p(r) p(r2) This is a very general result which relates the positional correlations of electron density fluctuations in the system to the measured scattering profile in the absence of approximations. In cases where a discrete treatment is more appropriate, a particle 2 particle correlation function replaces the term, p (r), in Equation 38.68 Often the systems studied with scattering methods are composed of particles having an approximately uniform electron density, p,. These particles are suspended in a gel or solution characterized by another electron density, P2. Neglecting inhomogeneities in the charge density, the scattering is equivalent to that from a system of particles with density Ap = p, P2 superimposed on a background of density P2. There will be no observable scattering from the background in the relevant scattering range. The problem is therefore reduced to the much simpler case of calculating the scattering from particles of density Ap suspended in a vacuum.64 The scattering amplitude for a system of N identical particles can be written N A(Q) = fp(r) exp( iQ( R + r )) dV (39) k=1 N = F(Q) exp(iQ.Rk) k=1 F(Q) = fp(r) exp(iQr) dV Here Rk is a vector extending from the origin to the center of symmetry of the kth particle and p(r) is the electron density distribution of a single particle about its center of symmetry. The quantity, F(Q), called the form factor for the particle, contains all information regarding intraparticle correlations. The structure factor, S(Q), is the product of the scattered amplitude, A, with its complex conjugate and is a weighted sum of the phase factors corresponding to the center of mass positions of all of the particles. S(Q) therefore contains information concerning interparticle correlations. The scattered intensity is proportional to S(Q) which can be written : I(Q) c S(Q) = F(Q) F*(Q) cos( Q.( R, R) ) (310) k After averaging over all particle orientations and separating those terms where j=k, this expression ( Equation 310) for S(Q) becomes S(Q) = N< IF(Q)2 > + NI< F(Q) >12 C cos( Q(RkRJ)) (311) kj The pair correlation function, P(r), introduced by Zernicke and Prins69 represents the probability that two particles are separated by a distance r. In the case of an isotropic distribution of particle positions and orientations, this can be used to evaluate the double summation in Equation 311. The resulting form for S(Q) is 00 < F(Q) >2 sin(Qr) S(Q) = N < F2(Q) > where v, is the average volume available per particle in the system. As v, decreases, interparticle interference effects become more pronounced. With v, large S(Q) is dominated by the first term in Equation 312. In this case, the observable scattering is just that of a single particle multiplied by the number of particles in the system. This is the limit typically explored in SAXS and the detailed properties of the scattered intensity in this regime will now be discussed. Small Angle Xray Scattering For particles with dimensions on the order of hundreds of angstroms and with a typical incident xray wavelength of 1.5 A, the majority of scattering features appear in the range from 0 to 2 degrees. This is typical of block copolymer solutions. Assuming a dilute, isotropic system of particles, the scattered intensity can be calculated from Equations 38. In all but the simplest cases this must be done numerically. For the important case of a hard sphere, the structure factor, first derived by Rayleigh,70 is S= Ap )2 V 3 sin(QR) QR cos( QR)2 ere (Q12 ( )2 V2 (QR) (313) where V is the volume and R the radius of the sphere. A plot of this function for three radii is shown on a linear and on a log scales in Figure 32. For the larger radius, the scattering function decays more rapidly reflecting the inverse relation between length scale and scattering vector. This function exhibits well defined maxima at the locations QR = 5.77, 9.10, 12.32, etc. from which the sphere radius can be determined. A more general method for determining the average size of particles exhibiting small anisotropy was developed by Guinier.71 In the limit of very small angles, Equation 0 4j 0 0 L E 0 0 L O 0 O LL) 0) 0 U Ou t LL O U L UL) 10 1 1 102 103 10 105 L 10.00 0.00 1.00 0.80 0.60 0.40 0.20 0.00  0.00 0.02 0.04 0.06 0.08 Scattering Vector 0.13 A1 A Figure 32 The scattering form factors for and ideal spheres having radii: 80, 100, and 120 A. 3.02 0.04 0.06 0.08 Scattering Vector 1 Scattering Vector ( A ) 38 reduces to 2< S(Q) = ( Ap )2 Vexp (314) fp(r) r2 dV where Rg = the radius of gyration = M Here the integral is taken over a single particle and M is the total mass of the particle. Equation 314 shows that if the log of the scattered intensity is plotted versus the square of the scattering vector ( called a Guinier plot), the slope of the resulting line will be proportional to the square of the radius of gyration in the limit of very small angles. Thus if the shape of the particle is known, the parameters describing its size may be determined from the radius of gyration as obtained from the scattering profile. For example, the radius of a sphere is related to its radius of gyration by Rg = [3/5 R. The Guinier approximation breaks down for severely anisotropic particles (i. e. plates or rods) and is invalid in the presence of interparticle interference effects. In practice, all of the particles do not have identical radii. Rather, there is a distribution of radii centered about a mean. In the case of block copolymers this effect arises from the fact that the polymer chains themselves are polydisperse. For example the scattering which results from a system of spheres characterized by a gaussian size distribution, having a mean radius of R0 = 100 A, and a standard deviations, a = 1, 10, and 20A are plotted in Figure 33. The number of particles, N, having radius, R, is given by N(R)= exp R)2 (315) and the total structure factor is the superposition of the structure factors from each particle in the distribution, S(Q) = fN(R) S pbe(Q,R) dR (316) 1.00 0.80 0.60 0.40 0.20 L 0 4a 0 0 LL E L 0 LL 0 0 c L_ /) 0 I, 0 LL E 0 L 0 CL c 100 10 102 103 10 104 1 0.00 0.02 0.04 0.06 0.08 0.10 Scattering Vector ( A ) 0.02 0.04 0.06 0.08 0.10 Scattering Vector ( A ') A plot showing the effect of polydispersity on the scattering form factor for a gaussian distribution of spherical particles having a width parameter, a = 1, 10, and 20 A. 0.00 1 0.00 Figure 33 The major effect of polydispersity is a weakening of the scattering minima which is more pronounced with larger o. Typically, the interface between the particles and background is of finite width. For block copolymers this region is where the two covalently joined blocks meet. If these junction points form a perfect surface, the interface is ideal. It is often assumed that the locations of these junction points are described by a gaussian distribution of width, This leads to a density profile p(r) = Ap 1 erf[ (317) The structure factor calculated from Equations 36 and 317, incorporating the effect of the interfacial thickness, 4, are shown in Figure 34. The spheres in this calculation have a mean radius of 100 A and values of = 1, 10, and 20 A. There is relatively little perturbation of the profile at very low Q, but the scattering at higher Q is greatly suppressed for large values of 4. In addition slight shifts in the locations of the scattering minima are observed. At large values of the scattering vector, a different limit is found. In the high Q region, the expression for the structure factor takes on a much simplified form 27n( Ap )2 S(Q) S (318) In this equation, Sa is the surface area of the scattering particle. This power law dependence is called Porod's Law.72 The radius of gyration and the surface area of the particles can be determined from the limiting forms given by Eqs. 314 and 318 and are very useful in deducing the shape and dimensions of the particle. If the surface of the particle is of finite thickness the scattering at high Q will exhibit systematic deviations from Porod's Law which depend on the interfacial thickness 4. In a method developed by Vonk73 and Ruland,74 the effect of finite 4 is taken into account by considering the electron density of the particle. p(r), as the convolution of an ideal electron density profile, p,(r), and a smoothing function, h(r). Since the scattering 0 0 E L 0 O 0 . O L.. 4 U 0 i E L Q. 0 U U) 1.00  0.80 0.60 0.40 0.20 0.00 0.00 10 10  10. ic2 102 o3 4 10  104 0.00 Scattering Vector The modifications to the scattered profile which result from spheres having imperfect boundaries are illustrated in these plots for a system of spheres having a radius of 100 A and values of x = 1, 10, and 20 A as described in the text. 0.02 0.04 0.06 0.08 0.10 Scattering Vector ( A Scattering Vector ( A ) 0.02 0.04 0.06 0.08 0.10 Figure 34 form factor, F(Q), is the Fourier transform of the electron density, p(r), it can be expressed as the product of the form factor for the ideal particle and the Fourier transform of h(r), Fh(Q). As described previously, h(r) is often approximated by a Gaussian having a width parameter equal to In this case the total structure factor takes the form S(Q) C Q4 exp( 4 X2 2 Q2) (319) Consequently, a plot of Q4 I(Q) vs Q2 yields the parameter, Application of this method to the study of block copolymers is discussed in Chapter 4. The above discussion was based on the assumption of widely spaced scattering centers so that the total scattering is the sum of the individual contributions. Now, the effect of a close packing of particles and interparticle interference will be discussed. Scattering From Liquids In the event that interparticle interference is present the scattering is given by Equation 312. In general, the pair correlation function, P(r), depends on the average volume per particle, v1, the temperature, T, and the interaction potential, Q(r). c(r) is the two particle potential, i. e. the potential energy of a system of two particles separated by a distance r. The determination of P(r) from the interaction potential is an extremely difficult problem which has been extensively studied.7577 The approach developed by Born and Green76 assumes a modified MaxwellBoltzman dependence of P(r) on Q(r). Their final result for the liquid structure factor is S(Q) = N < F(Q)2 > + < F(Q) >2 (2)3l(Q) 1 (320) where the function 3(Q) is defined as 0 Here e is a constant of the system which is on the order of unity. As an illustration of the above form, Equation 325 is evaluated for the hard sphere potential : I(r) = 0 r The resulting profiles for four values of where vo is the particle volume are plotted in Figure 35. Interference effects show up in the form of a reduction in intensity at very low Q, and an enhancement at intermediate Q (Q = 0.03 A, ). The result is a broad maximum which grows with increasing concentration. Scattering at higher Q ( Q > 0.045 A' ) remains mostly unaffected with the exception of a subtle shift in the position of the primary spherical scattering maximum (Q = 0.06 A ). The Van der Waals force is the origin for the interaction commonly occurring in many systems. The potential describing this interaction is well approximated by the LennardJones ( LJ) form : (](r) = (o 12 (323) This potential along with Equations 320 and 321 was successfully used by Fournet78'79 to predict the scattering from liquid and gaseous argon. Qualitatively, the scattering curves are very similar to those of Figure 35. One distinguishing feature is a sharp upturn in the scattering at small angles resulting from the long range of the LJ potential. In addition, a secondary interparticle interference maximum appears on the high Q side of the primary interference maximum. From the above discussion, it is apparent that the scattering at high Q in a concentrated system shows little deviation from single particle scattering. At lower Q, strong oscillations about the single particle scattering become evident and take the shape of broad peaks in the spectrum. In the case where the free energy favors a crystalline 49 The Hard Sphere 101 0 O 0 U 0 CO O" J Figure 35 0 10 B C D ff 10 E 102 0.00 0.00 0.02 0.04 0.06 0.08 Sctterin Vec1or ( A Scattering Vector ( A ) ".10 Model scattering profiles are shown for systems of spheres interacting via the hard core potential at varying particle concentrations, vl/vo. Model arrangement of particles, the assumption of an isotropic distribution of particles is no longer valid. The scattering from ordered structures will now be discussed. Scattering From Ordered Structures A crystal is a three dimensional periodic structure.0 The repeat unit for the structure is called the unit cell. The lattice is constructed by translating the unit cell by linear combinations of the basis vectors ( designated a,, a2, a3 ). A structure constructed in this way can be considered to be composed of planes of particles as is shown in Figure 36. The scattering amplitude from such a structure will be determined by the phase difference between waves diffracted from adjacent planes. If the path difference is equal to an integer number of wavelengths, there will be constructive interference otherwise the scattering amplitude will be zero. This is a qualitative statement of Bragg's Law62 which in its most common form is mX = 2d sin( 20/2) (324) Here d designates the distance between adjacent planes, X the wavelength of the scattered waves, and m the order of scattering. In general, adjacent planes are separated by al a2 a3 h k and in the three crystallographic directions. The integers (h,k,l) are called the Miller indices and are used to specify crystallographic planes. Calculating the distance between and the orientation of every group of crystallographic planes is a formidable task. The concept of a reciprocal lattice is useful in dealing with this type of calculation.80 In this theoretical construction three reciprocal lattice vectors are defined 2( a2 x a3 ) 2x( a3 x a ) 27( aI x a2 ) a,* a2xa3 a *a2 xa3' a a, axa3 An illustration of Bragg scattering from a crystal lattice. Figure 36 From vector algebra, one is able to show that the Bragg condition is equivalent to the requirement that the scattering vector, Q, be a linear combination of the reciprocal lattice vectors: S= hb1 + kb2 + lb3 (326) Within this construction, Bragg peaks are easily identified and designated by the Miller indices h, k, and 1. For Bragg's law to be satisfied, the angle of incidence relative to the set of scattering planes must equal the angle of reflection. This implies that Q is required to be perpendicular to these planes and, for an orthorhombic system, has magnitude Q = = 2r h + +2 1 )2 (327) Thus, constructive interference from a perfect crystal will only occur at specific values of the scattering angle, 28, and for specific orientations of the crystal relative to the incident beam. The Effect of Finite Size A real crystal is generally composed of a large number of crystallites of finite size. The scattered intensity from a single finitesized crystallite having n particles in the unit cell, and N,, N2, and N3 unit cells in the x, y, and z directions can be calculated62 and is proportional to the structure factor S(Q), S(Q)= A A* sin2( N Q'al / 2) sin2( N2 Qa / 2) sin2( N3 Qa / 2) sin2( QaI /2) sin2( Qa / 2) sin2( Qa3 / 2) where A(Q) = F(Q) eiQri In the above expression the scattering amplitude, A(Q) is a function of the form factors, Fi(Q), for n particles at positions ri within the unit cell. This function is peaked when the following three conditions are satisfied: Qai = 2xh, Qa2 = 2xk, Qa3 = 2rl (329) or equivalently, when Q is a linear combination of the reciprocal lattice vectors. In deriving Eq. 328 it was assumed that the grains were perfect parallelpipedons of uniform size. In the limit of large N1, N2, and N3 and in real systems having a distribution of grain sizes and shapes, the scattering lineshape is typically well approximated by the gaussian, having an amplitude, A and a width, af. The Crystal Structure Factor From the above discussion it would appear that there exists a peak in intensity corresponding to all integer values of h,k, and 1. This is true for a simple cubic structure. However, for any structure having more than one particle in the unit cell, the structure factor vanishes for certain combinations of h,k, and 1. The vector, r, from Equation 333 which specifies the location of each particle within the unit cell can be written as ri= x, a + y, a2 + zi a3 (330) where xi, Yi, and zi are belong to the interval: [ 0, 1] The scattering amplitude can now be rewritten as A(Q) = 2Fi(Q) ei( hxi +y + ) (331) i=1 For a bodycentered cubic structure ( BCC), there are two particles in the unit cell having coordinates : x, = y, = z, = 0, and x2 = y2 = zz = 1/2. The scattering amplitude for identical particles is A(Q) = 0, h+k+1 odd (332) A(Q) = 2 F(Q) h+k+1 even So the crystal structure factor, SBCC(Q) = A A*, vanishes at those peak locations where h+k+l is odd, and at those locations where h+k+1 is even, the peak is present having an effective form factor equal to twice that of the constituent particles. In a similar way, the scattering amplitude can be determined for the facecentered cubic (FCC) structure, A(Q) = 0 h, k, 1 mixed (333) A(Q) = 4 F(Q) h, k, 1 unmixed Only those peaks denoted by indices that are all even or all odd will be present ( i. e. (111), (200), (311) etc.). The Effect of Particle Motion Until now it has been assumed that the particles are stationary on their lattice positions. In fact, the particles exhibit displacement oscillations about their equilibrium lattice positions. In conventional crystals, these motions are the thermal vibrations of the atoms. For the block copolymer system, thermal oscillations for the massive polystyrene domains suspended within the viscous polybutadiene matrix are negligible. In these materials, the displacements result from diffusive motion of the domains in the vicinity of their lattice positions. The change in the diffraction profile due to particle motion can be described as a modification of the form factor in those cases where the time characterizing particle motion is much shorter than observation times. At any time the position of the kth particle can be written Rky = Rk + B(t). In xray scattering we measure the temporal average (indicated by triangular brackets ) of the scattered intensity, which is proportional to the structure factor: S(Q) iQ(RkRI) iQ(S(t)(8(t)) S(Q)= Fk2(Q) e < e > (334) k If uk(t) and u,(t) are defined to be the amplitudes of oscillation in the direction parallel to Q, then equation 334 can be rewritten as S(Q) = S(Q) + Sf(Q) (335) 2 SI(Q) = > ^F,k(Q)e J IFi(Q)e Je k The second contribution in Equation 335, Sd, contains the cross term, < ukUl > which is vanishingly small unless the kth and Ith particles are very close. The result is a scattering component that is slowly varying with Q and of very low amplitude. This contribution to the scattering is called the thermal diffuse scattering and is approximately constant in the high Q region of the spectrum. The first term, SI, is identical to the result for the stationary finitesized crystal with the form factor replaced by an effective form factor Feff(Q) = F(Q) exp(  Q2< u2>) (336) The exponential in the above equation is called the DebyeWaller factor. The peak widths and positions are unchanged by displacement oscillations, but the peak amplitudes are modulated by this factor. The Effect of Strain In a typical experiment scattering occurs not from one, but from many crystallites. In general there exist random strain fields within the sample. The result is a collection of grains having a narrow distribution of lattice constants centered about the equilibrium value. If we define the strain, e = Aa/a, then the distribution of strains may be approximated by 1 e2 P(e) exp( ) (337) 2 OY 20, For simplicity we have assumed an isotropic gaussian distribution of strains and a cubic lattice. The magnitude of the scattering vector at the bragg peak position is related to the lattice constant via Eq. 327. The Bragg peak will be shifted by, AQ, if the lattice constant is stretched by Aa. The peak shift and strain are related in the following way: AQ = Q (/a) Aa = Q e (338) so that the distribution of peak offsets is given by 1 (AQ)2 P(AQ)= exp( 2 ) OQ = Q (339) V2itOQ 20Q The scattered intensity is the convolution of this function with the profile for a strainfree system. The scattering profile for a strained system in this approximation is the convolution of two gaussians which is itself a gaussian having a width equal to the sum of the individual widths added in quadrature: (QQo)2 2 I(Q) = A (f/o) exp( 2a ), O = ( (o+ (a.Q)2 )1 (340) So the effect of strain is to broaden the peak by aoQ with an accompanying decrease in the Bragg amplitude so that the integrated intensity of the peak is conserved. It is evident that strain broadening is much more pronounced in the higher order peaks. The Powder Pattern If the orientation of the crystallites is random, the sample is called a powder. The scattering will then be a function of the magnitude of Q only. Assuming a Gaussian lineshape for the nonpowder averaged Bragg reflection, the powder averaged structure factor can be calculated (Q) = (QQ0)21 Q (27 2)3/2 < exp 22 > (341) Here the brackets indicate an average of the structure factor over all orientations of the Bragg center, Q0, relative to the scattering vector, Q. The form factor which has been incorporated into A(Q) is assumed to be isotropic. Without loss of generality, Q is chosen to lie in the z direction: Q = Q Z, Qo = Qosin(0)cos(()) + Qosin(0)sin( )) + Q0sin(0)2 (342) Here 0 and 4 are the conventional spherical coordinates. The average over 4 and 0 yields Ao (QQo)2 (Q+Q)2 S(Q)= Q [exp Q +exp [ ) (343) The second exponential is negligible for all positive values of Q. There is an additional factor called the multiplicity, mnh which arises from the fact that Bragg peaks having different indices h, k, 1 appear in the same location in the powder spectrum. The intensity is proportional to the degeneracy of the Bragg reflection. As examples, the multiplicities of several peaks for a simple cubic lattice are listed below (hkl)= (100) m= 6 (344) (110) m= 12 (321) m= 48 It is possible to identify different crystal structures by comparing the relative amplitudes of the observed Bragg peaks due to differences in the multiplicities of the different structures. This will be discussed in greater detail in Chapter 6. If the effects of temperature, strain, and powderaveraging are included, the resulting structure factor for a single Bragg peak is S(Q) = Ah (Q) exp[( 02 (345) mlw e F(Q)2 exp( where Ag(Q) = Ao ,) QQ 4[2(Q) QQ% and o(Q) =[ o + ( oQ)2 ' The application of these results to the block copolymer SAXS spectra will be discussed in Chapter 6 following a review of the block copolymer literature in Chapter 4 and a discussion of the experimental apparatus in Chapter 5. CHAPTER 4 BACKGROUND AND LITERATURE REVIEW OF BLOCK COPOLYMERS The polymer molecule is a long flexible chain composed of covalently bonded repeat units called monomers. Polymeric materials have long been of interest to chemists and materials scientists due largely to their advantageous mechanical properties and were only much later recognized as an excellent system for the study of physics. Major experimental tools that have proved to be extremely useful in recent studies of polymer structure, morphology, and dynamical behavior include small angle neutron scattering ( SANS ) and synchrotron xray diffraction. Outstanding theoretical advancements have been made, especially through the application of renormalization group ideas to the study of chain conformations, due primarily to the original work by de Gennes in this field.22 PolymersGeneral Background The size of a single polymer molecule is typically described by the polymerization index, N, N= (41) MO where M is the mass of the entire chain and Mo is the mass of a single monomer. Generally, a polymer sample is composed of a distribution of chains having varying total masses. The mass distribution can be characterized by three parameters: the number and weight average molecular weights, M. and M,, respectively, and the polydispersity index, n. The number average molecular weight is defined as IN. M. Mn =, (42) Mf. XN. Here Ni is the number of polymer chains of mass, Mi, in the distribution. Similarly, the weight average molecular weight is given by Ni, Mi2 M. = (43) The polydispersity is a measure of the width of the mass distribution and is defined to be the ratio of Mw and Mn. It is easily related to the standard deviation of the distribution, o, through the equation S= (< M2 > < M >2)12 = M, (nl)l2 (44) Together the above parameters describe the mass distribution of a collection of polymer chains. The size of individual chains is described by the endtoend distance, r. If each monomer is of length I then the rms value of r can be determined by considering a polymer chain as a random walk containing N steps of length a. It is well known2 that for an unconstrained or "ideal" random walk the result is Within this model, the entropy of a single polymer can be calculated and at fixed end to end distance is 3r2 S(r) = S(0) 2 N (46) From the free energy relation, F=UTS, it is evident that the chain behaves as a Hooke's Law spring, when the entropy term dominates the free energy. The effective spring constant increases with increasing temperature. This illustrates the physical basis for the contraction of polymer materials which occurs upon heating. The above calculation is flawed in that chain conformations are allowed in which the chain "loops back" on itself so that two or more monomers occupy the same physical space. Calculations which do not allow these conformations are called self avoiding walks ( SAWs ) and, for dilute polymer solutions, yield a result similar to that for the ideal random walk: Here v is called the Flory exponent. Theoretical and experimental studies show v=3/s5.s In the concentrated limit, chains can no longer be considered as independent random walks and the result, although much more difficult to obtain22 is the same as that for the ideal random walk, Equation 45. Polymer chains are therefore swollen in dilute solutions containing a good solvent and exhibit ideal ( i. e. random walk ) conformations in the limit of a concentrated polymer solution or a pure polymer melt. The radius of gyration, R,, is a parameter conveniently measured in both light scattering and small angle xray scattering experiments and is defined as Mr? 1/2 Rg I (48) For an ideal random walk the radius of gyration is equal to one sixth of the rms endto end distance.82 The Glass and Melting Transitions Many polymer systems exhibit a melting temperature, T., above which the molecules are in constant relative motion. Below Tm, the polymer chains crystallize by forming helical or layered structures. In many cases, "bulky" monomers prevent efficient stacking of the polymer chains so that crystallization is prevented. These materials exhibit a glass transition temperature, Tg. Below T,, relative motion between chains is hindered by entanglement effects. Often small molecules called plasticizerss" are added to the material to increase flow by providing lubrication and lowering T,. Above the glass or melting temperatures and in the absence of chemical crosslinking, polymer molecules are constantly moving relative to one another. For a long time, the mode of interdiffusion for polymer molecules was not well understood. Presently, this process has been clarified and is described by the reputation model developed by de Gennes.22 Within this model, polymer chains move by "worming" their way through an open ended tube in which they are partially encased. The mean time required for a single chain to reptate through a length of tube equivalent to its own length is called the terminal time, ,, and the following dependence on the polymerization index, N, has been observed:22 S NX x = 3 3.3 (49) The translational diffusion coefficient, D,, is given by 2 D N2 (410) assuming gaussian chain statistics. Phase Separation in Polymer Blends Many of the interesting and useful properties of polymer materials result from segregated structures in polymer mixtures where the component polymers are incompatible. In this section, the phase separation process in polymer blends is discussed as an introduction to the more complex block copolymer case. Many of the features of polymer phase separation are elucidated in a mean field treatment developed by Flory21 and Huggins.83 Within this model, the free energy of a mixture of polymers A and B is expressed in terms of the FloryHuggins interaction parameter, x: X=k (EAB( EAA+EBB) (411) Here Exy is the energy corresponding to an XY nearestneighbor configuration and kBT is the thermal energy. The polymer system is typically assumed to be incompressible84 so that the volume fractions of A and B monomers can be written: ^A = ( and (g = 10. The free energy for the mixture, including entropy of mixing terms,22 is I11 F=kBTN lnc +kBTNl n(l0)+X (lI) (412) Here and are the A and B chain concentrations, respectively, in dimensionless NA B units. It is trivial22 to construct the phase diagram for this system including both the coexistence and spinodal curves. The result for the symmetric case is very similar in form to the phase diagram illustrated in Figure 21. From the phase diagram it is evident that at temperatures above the coexistence curve the equilibrium state of the system is a homogeneous mixture. Below the coexistence curve, the equilibrium state consists of phase separated A and Brich regions. As described in Chapter 2, many interesting kinetic effects can be observed in a system exhibiting this type of phase diagram. Specifically, upon lowering the temperature through the coexistence temperature to a final value in the metastable region, there exists an interfacial energy associated with formation of A or Brich droplets. The mechanism of transformation is nucleation and growth. When the final temperature is in the unstable region of Figure 21, the interfacial energy disappears and transformation occurs through spinodal decomposition. However, as mentioned in Chapter 2, a sharp spinodal line is an artifact of mean field theories such as the FloryHuggins model and therefore its true physical significance is unclear. Nucleation and spinodal decomposition have been observed in studies of phase separation in blends of polystyrene and poly( vinyl methyl ether ) by Nishi et al.85 They used light transmission, optical microscopy, and NMR techniques. They were able to identify a range for the spinodal point above and below which morphological structures corresponding to nucleation and spinodal decomposition, respectively, were observed. It is expected that qualitatively different behavior should be observed for phase separation in polymer blends in comparison to binary alloys since diffusion in polymer systems is believed to occur through the reputation mechanism. Reptation yields a chain mobility which is strongly dependent on the reciprocal space vector, Q, and, as a result, alters the equations of motion for the system. Nevertheless, Nishi's group found that early time spinodal decomposition behavior agrees well with Cahn's linearized model of spinodal decomposition,86 a result which is expected in the presence of longrange interactions, but never observed in metals. McMaster87 also observed a spinodal point in his light scattering study of polymer blends. In addition, the coarsening behavior observed following quenches into the metastable temperature regime, was consistent with the LifshitzSlyozov theory55 described in Chapter 2. However, coarsening following quenches into the unstable regime occurred much more rapidly. McMaster accounted for this behavior, invoking a viscous flow mechanism based on models developed by Tomotika.88 Much work has yet to be done on phase separation in polymer blends, but present results indicate that studies of these systems, because of their slow molecular diffusion, can yield valuable information toward understanding kinetic effects.' Block Copolvmers Bulk Properties As introduced in Chapter 1, the block copolymer ( BCP) is composed of two or more chemically distinct polymer chains joined by a covalent bond. When the block chains are incompatible, phase separation similar to that occurring in polymer blends takes place. The fundamental properties of bulk block copolymers are summarized in a review article by Hashimoto et. al.84 As described in Chapter 1, block copolymers exhibit a dissolution temperature, Td, below which microphase separation transition ( MST ) results in the lamellar, cylindrical, and spherical microstuctures shown in Figure 12 and above which the system is homogeneous. The morphology of the microstucture has been found both experimentally23 and theoretically89 to depend only on the fractional composition, f, of the block copolymer and is believed to be independent of total polymerization index, N. In addition, the microphase separated domains are known to form macrolattices: a lamellar arrangement, a 2D hexagonal closed packed array of cylinders and a cubic lattice of spherical domains. Early theoretical work toward understanding this process was performed by Meier.90 The driving force for phase separation in a BCP is the repulsive interaction between the two types of monomers. Since the polymer system is highly incompressible,28 only those domain configurations which fill all space are allowed. If the size of the domains exceed the sums of the radii of gyration of the block chains composing the domain, then that chain must be stretched to satisfy the requirement of uniform space filling. This results in a loss of conformational entropy and therefore limits the size of the phase separated domains. There is an additional entropy loss due to confinement of the junction points (the location of the covalent bond between block chains) to the interfacial region at the surface of the domain. Meier first identified these three factors which determine the equilibrium morphology and dimensions of the system. In addition, Meier recognized that the domains themselves interact through a potential of entropic origin which in turn is responsible for the macrolattices which the microdomains are known to form. The physical basis for this interaction can be understood in the following way. Consider an AB diblock copolymer which forms spherical domains of the A block in a B matrix. If two A spheres are too far apart, then a density deficiency is created between them which must be filled by stretching B chains. In this case, the free energy is increased due to the loss of conformational entropy of the stretched B chains. Similarly, if two A domains are too close, the volume available to the B chains between them is too small, again resulting in an increased free energy due to the loss of conformational entropy. As discussed earlier in this chapter, the contribution to the free energy from stretching or compressing a polymer chain is quasielastic in form. One of the first attempts to predict the dimensions of the equilibrium structure is that of Helfand and Wasserman.91 Within their model, the structure is composed of chains individually constructed from self avoiding random walks confined to the appropriate domain space. In the limit of strong segregation for the lamellar morphology, they found the following scaling relation between the polymerization index, N, and the microdomain periodicity, d: d N, 0 = 0.636 (413) Similar results for the value of 0 were obtained for the cylindrical and spherical morphologies.9293 The result given in Equation 413 was supported by studies of the polystyrene polyisoprene system by Hashimoto et al.94,95 They cast polymer films from solutions having fractional compositions that yield the lamellar and spherical structures at varying total degrees of polymerization. The films were studied with the small angle xray scattering technique. From the SAXS profiles, the periodicities of the lamellar structures, D, were calculated as well as the lattice constants and spherical radii, R, of the spherical structures. In addition, Hashimoto's group studied the systematic deviations from Porod's Law in the high Q region of the scattered profiles ( discussed in Chapter 3) to estimate the thickness of the interfacial region, 4, separating polystyrene and polyisoprene domains. Their results indicate the following scaling relations for the lamellar samples: d Na (414) SN For the spherical structure, the same relations were found to hold and, in addition, the following empirical relationships were observed: R N4 (415) p, N'13 In these equations, pj is the density of junction points on the surface of the spherical microdomains calculated from the sphere radius, the bulk density of polystyrene, and the molecular weight of a single polystyrene block. For both morphologies, the periodicity scales with molecular weight in fair agreement with theoretical predictions.93 The exponent in this relation is greater than the exponent of 1/2 predicted for the radius of gyration, R, of the chain in a homogeneous melt.22 This indicates that the individual chains are stretched in the direction perpendicular to the domain surface. A more recent study confirms this assertion. Hasegawa et. al.9 studied lamellar polystyrenepolyisoprene using small angle neutron scattering ( SANS ). By deuterating the polystyrene block of the polymer, they were able to determine the radius of gyration of these chains from the SANS profile. In their solvent cast films, the lamellae preferentially lie parallel to the film surface. By scattering from the film edge and surface they could measure R, both parallel and perpendicular to the domain surface, respectively. Their results indicate an elongation of the chains by 60% perpendicular to the lamellar surface and a contraction of 30% in each of the parallel directions. A vast amount of work on similar systems also supports these scaling relations for lamellar,25.97,98.99. '0101 cylindrical, and sphericallo0lo02z,03,104.l'1 microdomain morphologies in the strong segregation limit. Leibler89 presented a Landau mean field35 analysis of the MST occurring in an AB diblock copolymer having a fractional composition f, a FloryHuggins interaction parameter X, and a total polymerization index N. The order parameter Nf(r), was defined to be the local density deviation of the A monomer, WA(r), from its mean value in the homogeneous state; W(r) = < yA(r) f >. The static structure factor, S(Q), is defined as S(Q) = f exp( i Qr) < AA(r) AA(O) > d3r (416) Leibler expands the free energy in powers of the order parameter. Using the random phase approximation,22 S(Q) and the free energy are evaluated to fourth order in W(Q) ( the fourier transform of V(i) ). By assuming that the minimized free energy near the transition is dominated by fluctuations with Q = Q* (Q* Rg ), Leibler is able to evaluate the coexistence and spinodal curves in the (XN) f plane and determine the limits of stability of the different polymer morphologies. The stable morphology is found to depend not only on the fractional composition, f (as shown in figure 12 ), but also on the temperature. Such transitions, i. e.. from spherical to cylindrical to lamellar, with changing temperature were later observed experimentally.23 In addition, Leibler argues that the stable macrolattice for the cylindrical morphology is the two dimensional hexagonal closepacked structure (HCP) and for spherical domains is the BCC structure. There have since been reports of a variety of different macrolattices ( BCC as well as FCC and SC) observed for the spherical domains.'06 The above formulation yields a scaling relation between the periodicity, d = Q*', and the polymerization index, N, d N1I (417) for the weak segregation limit (where the density profile is a varies smoothly ), distinct from the relation derived for the strong segregation limit, where well defined interfaces exist ( Equations 413 and 414 ). Recent support has been given to this scaling relation by Chakrabarti et. al.107 who performed Monte Carlo simulations using a two dimensional lattice model composed of restricted selfavoiding random walks. Their results and experimental measurements by Shibayama et al.108 also yield the relation given in Equation 417. Ohta and Kawasaki'09 extended Leibler's theory by including the effects of the longrange elastic interaction. Using the random phase approximation in the strong segregation limit they obtain results similar to those of Leibler, and calculate the correct scaling relation for this limit (Equation 413). More recently, Frederickson and Helfand10 have attempted to improve upon Leibler's theory by including the effects of concentration fluctuations in the self consistent Hartree approximation."1 Their results are similar to those of Leibler, but they find the transition temperature to be no longer linear in N, but proportional to N1+I, where (3 is small. The differences between these predictions and those of Leibler are, however, small and have not yet been resolved experimentally. In addition to the three commonly observed morphologies: lamellar, cylindrical, and spherical, a new morphology called the ordered bicontinuous doublediamond (OBDD) structure has recently been observed2327 in the polystyrenepolyisoprene diblock copolymer. The structure is composed of tetrapod domains probably residing on the doublediamond lattice. This structure was observed in SAXS and electron microscopy studies when the volume fraction of polystyrene is in the narrow range between 0.62 and 0.66. The OBDD morphology appears between the cylindrical and lamellar morphologies in the accepted morphology scheme( Figure 12). This discovery shows the possibility of finding more interesting and technologically useful morphologies. Bulk Block Copolymers Kinetic Studies The theory of the kinetics of the MST is not well developed at the present time. Kawasaki and Sekimotol12 present a complex model of this process within the reputation model of polymer diffusion. These results are, however, still very preliminary. Oono and Bahiana'13 employed a cell dynamical model to describe the MST in a symmetric diblock copolymer in two dimensions. The partial differential equation used to describe the time development of the order parameter was a modification of the Cahn Hilliard equation.' The two dimensional patterns resulting from their computations accurately reproduced those observed in electron microscopy studies.84 Using the results of their computations, dimensional analysis, and comparisons to equilibrium theories 89.109 they obtain a relation between the exponent, 0, describing the equilibrium periodicity of the microphase structure (Equation 413) and the exponent, 4, describing growth of the microdomains during spinodal decomposition: Lto (418) That relation is 8 = 2) (419) This implies that in the strong segregation limit, where 0 is believed to be 2/3 for the lamellar morphology, f = 1/3, in agreement with the LifshitzSlyozov result described in Chapter 2. Block Copolymer Solutions The first observation of an ordered structure in block copolymer solutions occurred in 1966 by Vanzo.114 In studies of the optical reflectance of diblock copolymer solutions as a function of increasing polymer concentration, C, a critical concentration, C*, was found at which point a sharp increase in reflectance was observed. In addition, above C the peak in optical reflectance moved to larger wavelength with increasing C. From these data Vanzo postulated the presence of an ordered structure at concentrations above C with an identity period that increases with increasing polymer content. Early work by Sadron and Gallot25 helped to clarify these effects. They studied a polystyrenepolybutadiene diblock copolymer mixed with styrene monomer, a selective solvent ( one in which only the styrene blocks are soluble). After the solutions were prepared they were illuminated with UV light to polymerize the styrene monomers. Through this "postpolymerization" technique the polymer solutions are converted into a gel for electron microscopy studies presumably without altering the microstructure of the solution. The resulting sample were stained with osmium tetroxide and studied with transmission electron microscopy (TEM). From these experiments, Sadron and Gallot identified two critical concentrations, C1" < C"'. Below C,*, the mixture was homogeneous. In the region between C1" and C2*, cylindrical polybutadiene domains were observed in a disordered arrangement. Above C2*, the cylindrical domains were observed in and ordered arrangement. From these results, it was inferred that C2* corresponds to the quantity C* observed by Vanzo. Sadron and Gallot25 and Skoulios97 also observed morphological transitions (i. e. from spherical to cylindrical, or cylindrical to lamellar microstructures ) with changing polymer concentration in solutions. Using the same "postpolymerization" technique they were also able to observe morphological transformations upon altering the fractional composition, f, of the polymer chain. Pico and Williamsn15 studied the ordering transition in triblock copolymer solutions using theological measurements. A transition from nonNewtonian to Newtonian viscous flow was observed upon increasing the polymer concentration above the measured critical value. The properties of BCP solutions have been systematically explored by many researchers, 03, 116.117 using a variety of techniques, but perhaps the most complete description of these complex systems is presented in a series of five papers by Hashimoto et al.28, 8,119,120.121 They performed theological and small angle scattering measurements of the equilibrium structure of diblock copolymers in both selective and nonselective solvents as a function of temperature and polymer concentration. A brief discussion of the equilibrium and kinetic results reported by Hashimoto et. al. is given here. Equilibrium Measurements The first two papers of this series28 118 report studies of the diblock copolymer polystyrene (PS )polybutadiene (PB ) in the selective solvent ntetradecane (C14) which dissolves the PB block. The BCP had a number average molecular weight, M. = 5.2 x 104 and a fractional composition, f = 0.30. The solutions studied had the following polymer concentrations: 8, 11, 20, 35, and 60 wt.% polymer. At these concentrations and at this fractional composition the MST results in spherical PS domains within a PB/C14 matrix. Xray scattering profiles were acquired at varying temperatures upon heating and cooling. The xray spectra were corrected for parasitic scattering and collimation errors resulting from instrumental resolution. In these experiments, Bragg peaks were evident in the low temperature spectra for all of the solutions in the relative locations Nl/, "f2, '3, and v4 after collimation correction. The lattice constants were determined from the first Bragg peak location and the average radius of the PS spheres was determined from the location of the first maximum in the spherical form factor as described in Chapter 3. It was speculated that the macrolattice of PS domains was simple cubic ( SC ) rather than bodycentered cubic ( BCC) from comparisons of the calculated and stoichiometric PS volume fractions for these structures. This type of calculation is described in detail in Chapter 6. As the temperature, T, for each solution was increased, the Bragg peak amplitude decreased continuously while the full width at half maximum ( FWHM ) remained unchanged. In addition, the calculated lattice constants decreased slowly with increasing temperature. At a polymer concentration dependent temperature To, the Bragg amplitude decreased abruptly, accompanied by an increase in the FWHM and a drop in the lattice constant. As the temperature was increased further above T,, the peak amplitude and lattice constant continued to decrease, while the width increased rapidly. Finally above T = Td, no further evidence of the peak structure remained. Hashimoto et al. associated the temperature To with the disordering of the macrolattice resulting in a "liquidlike" arrangement of the PS spheres. The temperature, Td, is identified as the dissolution temperature at which point the PS domains totally dissolve. In order to clarify these interpretation, theological measurements were made on the polymer solutions as a function of temperature. Two transitions in the mechanical properties were identified. Upon raising T above To, a transition from nonlinear plastic flow to linear nonNewtonian flow was observed in support of their identification of T,. Also, at T = Td, a transformation from linear nonNewtonian flow to linear Newtonian flow was seen in agreement with earlier studies15 identifying this behavior with the dissolution point. The authors identify a number of quantitative trends in the thermal behavior of the system. The dependence of the apparent lattice spacing, a, on temperature both above and below the ordering point were consistent with the following empirical relation above and below the discontinuity at T,: a T (420) The discontinuity in a at T = To is thought to be due to the change in structure occurring at this temperature. Above T,, the peak is no longer a Bragg peak, and therefore, determining a lattice constant from the peak position is erroneous. The decreasing trend of a with T is the result of two factors. First, the PB chains between the domains contract upon heating to increase their conformational entropy. Second, at higher temperatures the effective interaction, given by the FloryHuggins parameter, X 1/kBT, is reduced resulting in greater intermixing of the PS and PB chains which in turn causes a contraction of the lattice. The authors believe the second of these two is the more important effect at all temperatures. The Bragg widths below To appear to be independent of temperature. The authors offer an interesting explanation for this result. The domaindomain interaction in BCP solutions is assumed to be much the same as described by Meier for the pure polymer system, being quasielastic. The change in free energy, AF, due to small elongations, Aa, of the lattice constant from its equilibrium value can therefore be expressed as AF f(C, T) (Aa)2 (421) Here f(C,T) the spring constant of the PB chains, is a function of the polymer concentration, C, and temperature. The authors argue that if the distribution of strains satisfies the Boltzmann distribution, then the distribution of lattice constants is characterized by a gaussian form having a width parameter, o,, kBT a2 (CT)f(CT) (422) The "spring constant" for the PB chains is entropic in origin as described earlier in this chapter so that f ( C, T) kBT (423) Together 422 and 423 imply a strain distribution which is independent of temperature. Therefore the Bragg peak width is expected to be independent of temperature when strain is the dominant contributor to the width. The authors propose the following intuitive explanation for disordering of the macrolattice. As the temperature is increased and there is greater intermixing of the PS and PB chains, the PS blocks extend further into the interstitial region between the spheres. The PB chains are thus relaxed by intermixing and the effective potential well of equation 421 becomes more shallow until melting of the macrolattice results. When the demixing becomes pronounced, the domaindomain interaction is no longer purely entropic. The energetic contribution to the interaction is claimed to be the cause of the increasing trend in the peak width with temperature just below the dissolution temperature. Hashimoto et al. also observed several trends with increasing polymer concentration. Both To and Td increased with increasing concentration due to reduced screening of the PSPB interaction which results in a higher FloryHuggins interaction parameter, X. The measured lattice constants were consistent with the scaling relation a C'" (424) Again, the data extend significantly less than a decade in both a and C. The PS sphere radius, R, was seen to increase slowly with increasing C. This trend was said to be the result of a shifting in the relative weights of the energetic and entropic terms in the free energy. At higher concentrations, more chains are added to each domain increasing the sphere radius. As a result the PS chains are stretched and some conformational entropy is lost, but the interaction energy from the surface to volume ratio is reduced. TimeDependent Experiments As an extension of their earlier work on equilibrium properties, Hashimoto et. al. performed what appear to be the only kinetic studies of the block copolymer system to date.122,123.124 They again studied PS PB diblock copolymer in the selective solvent C14. The fractional composition was f = 0.30, and the molecular weight was Mn = 5.2 x 104. The polymer concentrations studied were 20, 25, 30, 35, and 40 wt.% polymer in solvent. They attempted to measure the polymer diffusion coefficient as a function of temperature by performing temperature jump experiments using the timedependent small angle xray scattering technique. These measurements are described in some detail to provide a comparison with the experiments reported in this thesis. The polymer solutions were initially held in equilibrium at room temperature. The temperature was then rapidly raised to a point above the dissolution point. With the thermodynamic driving force for the MST removed, it was thought that the system would relax to the homogeneous state through diffusion of the center of mass positions of the diblock chains. By studying the SAXS profile as a function of time, it was their goal to measure the diffusion constant, DC, for the diblock chains in solution. In fact, even at elevated temperatures, the interaction between PS and PB monomers will still affect the dissolution process. In addition, the diffusivity of the whole chain is a function of the diffusivities for each of the individual blocks. As a result, the measured quantity is an effective diffusivity, De. The polymer solutions were placed in an aluminum sample cell, which was sealed with mica windows, and held at room temperature. The sample cell was then rapidly transferred manually to a large copper heating block which was controlled at the desired final temperature. The sample temperature exhibited a sigmoidal variation with time and a time constant of 6 seconds. Placement of the cell in the heating block triggered the start of data acquisition. Sixtyfour consecutive scattered spectra were taken in real time following the temperature jump, each with an exposure time of 2 seconds. Due to poor counting statistics, temperature jumps to each final temperature were repeated 10 times and for each iteration, the spectra corresponding to the same time slice were summed. The resulting SAXS profiles exhibit a firstorder peak which decays rapidly with time following the temperature jump. A relation between the scattered intensity, I(Q,t), and Df. was derived involving three assumptions: an isotropic particle distribution, Fickian diffusion of the diblock chains, and no center of mass motion for the microdomains while they dissolve, i. e.. no macrolattice disordering. Their result is I(Q,t) = I(Q,t=0) e2 Q2 Dl t (425) From this expression, the intensity decays exponentially with time and so D.f can be determined from the slope of the semilog plot. The interpretation of these results is complicated by the speed with which transformation takes place. Much of the transition occurs nonisothermally. In order to account for these nonisothermal effects, a crude model for the time development of the scattered intensity was derived and is described by the following recursion relation: I(Q,til) = [ I(Q,) I(Q,T) ] e2 Q2 Ddf(T) At + I(Q,Tj) (426) Here the temperature jump is divided into time increments, At. The intensity at time, tj. is determined from the intensity at the previous time increment, tj. During each increment, the intensity moves toward the equilibrium intensity distribution, Ie corresponding to the temperature Tj which is the temperature of the sample at time tj. In order to further simplify the problem, D, is assumed to follow an Arrhenius temperature dependence: Dff = Do exp (AH ) (427) Their model for the temporal development of the intensity at any scattering vector, Q, is a function of the two parameters, Do, and the activation energy, AH,. The authors plot the logarithm of the scattered intensity at a specific scattering vector, Q*, near the firstorder maximum as a function of time following the quench. The data are generally nonexponential at early times becoming exponential at later times. This behavior is accurately reproduced in their model described above. The authors present two methods of extracting the desired parameters from the data. First, the diffusion constants Def are determined from the late time exponential region of the data and from a plot of Dff versus temperature, AH, and Do can be calculated. The second method, involves varying Do and AH. and individually matching the model ( Equation 426 ) to the measured data. The two methods gave similar numerical results for AHa and Do. The data were found to be consistent with the following relation: Do C'.75 (428) Results for the activation energy, AHa, show no consistent trend, but yield results for all concentrations in the vicinity of 10 kcal / mole. As can be seen from this brief discussion of the current literature, experimental studies of BCP transition phenomena are still quite limited. These systems are rich in their diverse microstructural morphologies and the transitions which occur between them 78 and the homogeneous melt. This complex behavior has been the stimulus for the series of experiments reported here. CHAPTER 5 THE POLYMER XRAY SCATTERING EXPERIMENT In this chapter, the experimental apparatus is described beginning with a brief overview of the data acquisition system. This is followed by a more detailed discussion of the major components of the apparatus: the polymer xray scattering furnace and the SAXS apparatus. The chapter is concluded with a discussion the experimental procedures employed in these studies. System Overview The objective of these studies was to observe structural changes in block copolymer solutions as a function of temperature and time by measuring the SAXS profile. The data acquisition system which was devised for this purpose is shown in Figure 51. The control system is centered around an IBM PC AT operating in the ASYST environment. The programs and routines written to control the data acquisition system are described in Appendix A. The computer controls the xray detector and polymer xray scattering furnace through four devices controlled via the General Purpose Interface Bus. These devices are a PAR 1461 detector controller, a Stanford Research Systems Model DG535 digital delay generator, a Micristar temperature controller, and a Keithley 197 digital multimeter. The PAR detector interface controls the PAR 1412 XR position sensitive xray detector. Upon receipt of commands from the computer, the detector interface relays those commands to the detector, then stores up to 512 acquired spectra for DMA transfer A schematic of the data acquisition system. Figure 51 to the computer at a later time. The silicon diode array detector is evacuated using a Leibold D8A twostage roughing pump and requires a cooling system ( A Fisher Model 900 Circulator ). The details of the SAXS apparatus including the detection system are discussed momentarily. The sample temperature is monitored and controlled by a Micristar temperature controller operating in the proportional mode of control. In this mode, the Micristar compares the sample temperature to the programmed setpoint and outputs a DC voltage between 0 and 5V which is proportional to the difference. This voltage is sent to a specially designed heating circuit ( described in Appendix B ) which in turn controls the current to heating resistors in the polymer xray scattering furnace. In this way, the sample temperature is controlled with a stability of + 0.3 oC. The details of the xray scattering furnace are discussed momentarily. The sample temperature as determined from the thermocouple voltage emitted by a thermocouple embedded in the sample is also monitored by a Linseis Model L4100 chart recorder and the Keithley multimeter. The timedependent measurements were performed by annealing the BCP solution above its dissolution point, Td, and then rapidly lowering the temperature to a fixed point below Td and observing the structural transition through the SAXS profile. This thermal quench is achieved by forcing a fixed amount of coolant rapidly through a quenching channel in the sample mount. The measured coolant was injected into the aperture of the quenching channel inlet to the furnace. Polypropylene tubing extending from a cannister of helium was then connected to the quenching channel inlet. A solenoid valve placed between the furnace and the helium was used to control the time and duration of the helium burst which was initiates the quench. The solenoid valve is controlled by the digital delay generator through a specially designed solenoid interface circuit, a schematic of which is shown in Appendix C. A description of the procedures performed in these experiments is given later in this chapter. Now, a more detailed discussion of the major components to the data acquisition system, the polymer furnace and the SAXS apparatus, is presented. The Polymer Xray Scattering Furnace The polymer quenching furnace is composed of three parts : the furnace body, the translational stage and the sample mount. The furnace body and the translational stage were originally designed for use in other experiments and were used here with relatively minor modifications. The sample mount, however, was specifically designed for these experiments and will therefore be described in slightly more detail. The Furnace Body The body of the furnace is essentially a vacuum chamber composed of a stainless steel base and an aluminum lid. These two parts are joined with an oring seal and a number of bolts arranged in an octagonal pattern. The furnace base as seen from above is shown in Figure 52. The lid is a cylindrical aluminum shell with two aluminum windows for entrance of the incident xray beam and exit of the scattered radiation. The windows are aluminum plates with 21/4" diameter holes. The window frames are bolted to the lid for easy replacement. The frames are sealed to the lid with orings. A number of window materials were tried and the two optimal materials were beryllium and Kapton. Kapton is the tradename for a polyimide material commonly distributed in sheets and used as substrates or vacuum windows. Beryllium is the standard window material used in wide angle xray scattering and offers the best percentage transmission ( 96.17% for a 5 mil sheet). However, it is a horrendous parasitic scatterer in the region of interest in these A top view of the polymer xray scattering furnace illustrating the relative placement of its various features described in the text. Figure 52 experiments. Parasitic scattering is defined as that unwanted component to the scattering profile which is a result of scattering from slits and window materials. This quantity was measured as the integrated number of counts per second above background. This quantity for the 5 mil beryllium sheet was 8.0 times larger than for the 5 mil Kapton sheet in the 0.05 to 1.0 degree range and 5.2 times larger in the 0.05 to 0.35 degree range. Although the percentage transmission was slightly less (transmission = 91.13%) for the Kapton, the reduction in parasitic scattering was much more important. The Kapton windows were sealed to the window frame with Omegabond epoxy and resin. The chamber was evacuated with an Edwards ETPG pumping station which incorporated a roughing pump and a turbomolecular pump. The ultimate pressure typically achieved under experimental conditions was between 1 and 10 millitorr. The pressure was much lower ( below the level which could be read from the Varian thermocouple gauge controller ) when the windows were replaced with solid aluminum plates. For this reason, it was concluded that the ultimate pressure was being limited by permeation through the Kapton windows. The vacuum pressure probably could have been improved with thicker Kapton windows, but the pressure was satisfactory for these measurements and further improvement was unnecessary. The furnace body has a number of vacuum feedthroughs. These are shown in figure 52 and include the following. An eightpin electrical feed through is used to pass current to the resistors which heat the sample. The feed through was a standard high vacuum flange which bolted to the side of the furnace and was sealed with teflon tape. Two 1/4" outer diameter, I/g" inner diameter copper tubes were passed through the furnace wall and were sealed with standard Swadgelock fittings. These tubes were used to pass coolant through the sample mount to initiate the thermal quench during the kinetic measurements. One feedthrough passes four thermocouple connections into the chamber: two chrome and two alumel. One chromel and one alumel connection are used to form the thermocouple junction which compose a single thermal sensor. Both sets of connections were used, one to monitor the sample temperature and one to monitor the sample mount temperature. A Varian Type 0531 thermocouple gauge head was attached in the location shown in figure 52 with teflon tape. The sensor was connected to a Varian 843 gauge controller. The chamber was evacuated through the 1/2 vacuum port shown in the Figure. Two additional high vacuum flanges are available on the scattering furnace, but are not being used at this time. The Translational Stage The base of the furnace body is cut into a dovetail shape. The base mates with a brass translational stage, the bottom of which is itself cut into the dovetail shape in the orthogonal direction. An additional brass plate mates with this piece, and by sliding the dovetail within its track, two independent translations are provided for centering the sample in the xray beam. The translational stages are mounted on a Huber Four Circle Diffractometer ( the four independent orientation angles were a little overkill in a small angle scattering experiment ) via the standard Huber Goniometer connecting ring. The connecting ring can be translated vertically. Along with those translations allowed by the brass translational stage, three degrees of freedom are provided for centering the sample in the beam. The connecting ring is typically adequate for small samples or sample chambers, but is inadequate for a furnace of this size. For this reason, appropriate spacers were chosen and the vertical translation of the connecting ring was set so that the furnace bottom rested on the spacers. Then four additional screws were used to further secure the furnace to the diffractometer. As stated previously, the furnace body and translational stage were originally designed for use in the Cu3Au experiments being performed in the laboratory and were used with minor modifications (i. e. installation of the quenching loop feedthoughs ). However, the sample holder was designed specifically for these experiments. Many iterations were required to optimize the performance of the sample mount. The development of the sample mount is described in Appendix D. Only the final version is described here. The Sample Mount The sample mount is composed of two parts, the heating unit and the sample cell both of which are shown in Figure 53. the heating unit was mounted in the furnace and remained there throughout the experiments, while the sample cell was replaced for each polymer solution. The sample cell is a 27 mm x 12 mm x 3 mm copper block. A slot of dimensions 2 mm x 5 mm was cut in the copper block to serve as the sample cavity. The polymer solutions were held in this cavity by two 2 mil Kapton windows. The Kapton windows were sealed using Omegabond high temperature epoxy. Two 1/16" stainless steel masks were bolted to the front and back of the sample cell to provide structural support to the seal. these masks were found to be essential in preserving the integrity of the seal at the elevated temperatures used in these experiments. The sample temperature was measured by inserting an Omegaclad Ktype thermocouple directly into the sample cavity through a 1/16" diameter hole. these thermocouples are composed of one 3 mil chromel and one 3 mil alumel wire extended parallel to each other and surrounded by a cylindrical stainless steel shell. This cylindrical sheath is packed with ceramic powder which prevents contact between either of the alloy wires and the outer shell. The metal coating and ceramic powder are sheared off at both ends of a 2" long section. At one end, the leads are welded together to form the thermocouple junction. at the other end, the wires are connected to the thermocouple The final sample mount is illustrated in a front and top view. Evident in this drawing are the quenching tubes, the heating resistors, the thermocouple, the stainless steel masks which support the Kapton windows, and the stainless steel brace which firmly clamps the sample cell into the sample mount. Figure 53 